Physics 2220 Fall 2010

George Williams

SECOND MIDTERM - REVIEW PROBLEMS

The last four problems are from last years second midterm. Solutions are available on the class web site.. There are

no solutions for 3, 22-24, 69, 111, 115, 141, 144. Gauss' Law problems, involving electric fields, from the last

review set are still fair game. Any problem involving gravity in the solution set is off limits.

1. (a) Given an electric potential of the form V = Bx y z, where B is a constant. Calculate the y component of3 2

the electric field.

(b) Calculate the equivalent capacitance between A and B of the

network shown. All capacitances are 2.00 pF.

(c) A 2.0 pf capacitor is connected in series with a 2.00 pF capacitor. What is the equivalent capacitance of

the combination?

(d) If a long metal cylinder of radius 1.00 mm has linear charge density of 3.25 × 10 C/m, what is the-6

electric field at its surface?

(e) A 30 pF capacitor is charged to 150 V. The battery is disconnected. It is then connected in parallel to a

40 pF capacitor. How much energy is lost?

2. (a) Calculate the electric force between two electrons 6.00 × 10 m apart.-10

(b) The electric potential in a region of space is given by V = -Ax y , where A is a constant. Calculate the y2 3

component of the electric field.

(c) Calculate the equivalent capacitance between A and B of the combination

below.

1 2 3C = 4.0 pF; C = 7.0 pF; C = 15.0 pF

(d) In a hydrogen atom, one electron orbits about one proton. If you assume a circular orbit (classical

model), and a distance between proton and electron of 0.50 × 10 m, calculate the velocity of the-10

electron.

(e) The energy stored in a capacitor is 1.75 J. If the voltage is 175 V, find the value of the capacitance.

3. (a) Calculate the equivalent capacitance of the series combination of 1200 pF and 1800 pF.

(b) A 12 volt battery is connected across a 100 :f capacitor. Find the charge on the capacitor.

(c) Find the value of g 1000 miles below the surface of the earth.

(d) Find the electric field 1.2 m away from a long cylinder of radius 0.10 m. The cylinder has a surface

charge density of 1.2 × 10 C/m .-6 2

(e) Find the electric field between the plates of a parallel plate capacitor if the area of the plates is 0.10 m ,2

the separation is 1.3 mm, the voltage across the plates is 130 V, and there is not dielectric between the

plates.

4. (a) Find the effective capacitance between points A and B of the

combination of capacitors shown.

(b) An electron at rest is accelerated through a potential difference of 172 V. Find its kinetic energy (in

Joules).

(c) Convert 3750 electron-volts to Joules.

(d) If the electric potential is given by V = Ax where A is a constant, find the x component of the electric2

field.

(e) Find the y component of the field due to the potential in (d).

5. (a) Calculate the potential energy of the square configuration of electric

charges given below.

(b) Find the potential at A before the charge is placed at A, but after the

other three charges are in place.

(c) Find the work to bring a charge of +q from infinity to point B at the

center of the square (with all 4 charges at the corners in place).

6. Given a spherical nonconductor with a uniform charge distribution of density D.

o o(a) Calculate the electric field at a point R /2, where R is the radius of the sphere.

o o(b) Calculate the total energy stored in the electric field between R /2 and R .

17. Given two thin concentric conducting spheres. One has a radius of R and the other a

1radius of 2R . There is a charge +Q on the inner and -Q on the outer sphere.

1(a) Find the electric field at a point P at R = 3/2 R .

1 1(b) Find the total energy stored in the electric field between R = 3/2 R and R = 2R .

8. Given two concentric, metal cylinders of length L and radii a and b. They are charged with

charges +Q and -Q as shown.

(a) Calculate the energy density at an arbitrary point P, a distance R from the center.

(b) Set up, but do not evaluate, an integral by which one could calculate the total energy

stored in this system.

9. (a) Calculate the effective capacitance between (a) and (b) of the network

given. All capacitances are 3.00 pF.

(b) Given an electric potential of the form V = Ax y z where A is a constant. Calculate the y component of4 7 2

the electric field.

(c) Three charges of equal magnitude |Q| = 4.65 :C, are the corners of an equilateral

triangle whose sides a = 4.25 cm. Calculate the electric potential, including sign,

at point P, at the midpoint between the two positive charges.

(d) A 3.75 pF capacitor is charged to 350 V. Calculate the energy stored in the capacitor.

(e) Calculate the capacitance of a parallel plate capacitor if the area of each plate is 3.75 cm , and the plate2

separation is 0.100 mm. The plates have a vacuum between them.

10. (a) Calculate the electric potential (in volts) at the center of a quarter circle of

charged wire if the charge density is 8 = 4.20 × 10 C/m and the radius is-9

15.0 cm.

(b) If the potential is in the form V = Ay z , calculate the x component of the electric field at the point2 3

(1,2,3).

(c) Calculate the electric potential difference needed to give an electron a velocity of 1.5 × 10 m/s. 5

(d) An electron is released from rest in a uniform electric field of 1.75 × 10 V/m. Calculate its velocity-6

after 1.5 seconds have passed.

(e) In moving a charge of *q* = 4.0 ×10 C from A to B, 6.5 × 10 J of work is done. Calculate the-12 -7

potential difference between A and B.

11. Given a semi-infinite line of charge of density +8 c/m. The line

begins at x = 0, and extends to infinity along the x axis. Calculate the

A Belectric potential difference between points A and B (V - V ). (Hint:

Instead of infinity, use a large value of x, and see what happens.)

12. (a) The electric potential difference between two charged metal

plates that are parallel is 127.0 V. Calculate the magnitude of the work to move an external small

charge of 7.25 :C from one plate to another. (The charge is moved slowly.)

(b) A capacitor of C = 17.2 pF is given a potential of 75.0 V. Calculate the charge on the capacitor.

(c) If the potential in a region of space is given by V = +A xy z volts, where A is a positive constant,2 3

calculate the y component of the electric field at the point x = +2.00 m, y = +3.00 m, z = +2.00 m.

(d) If all capacitors in the arrangement shown have the same value C,

calculate the effective capacitance between a and b.

(e) Calculate the capacitance of a parallel plate capacitor if the plates are squares with sides of 1.50 × 10-2

m, and the gap between plates is 0.275 mm.

13. Given the configuration shown.

1 2Q = +4.00 :C; Q = +7.00 :C

3Q = !5.00 :C; a = 3.00 × 10 m!3

(a) Calculate the electric potential, including sign, at point P.

(b) Find the work done by someone who brings a charge of +6.00 :C

from infinity to point P.

(c) Calculate the potential energy of the configuration of 4 charges

after the charge in (b) is brought to point P.

14. In the arrangement shown, switch A is closed for a while and then

opened. After A is open switch B is closed.

(a) Find the potential on each capacitor after B is closed.

(b) Calculate the charge on each capacitor after B is closed.

batt 1 2 3V = 175 V; C = 1.25 :F; C = 2.40 :F; C = 3.00 :F

15. (a) Calculate the electric potential at point P due to the four charges

shown.

(b) Calculate the electric potential at point S due to the four charges

shown.

16. A long cylindrical rod (non-conductor) has a uniform density of D C/m . 3

oThe cross section of the rod is circular with a radius of R .

(a) Calculate the electric field at an arbitrary interior point of the cylinder,

Aa distance R from the center, and far from either end.

o A(b) If the rod is negatively charged, state the sign of V(R ) - V(R ).

Explain clearly in words why this is the correct answer.

17. (a) Given an electric potential of the form V = Ax y z , calculate the y-component of the electric field at the2 3 4

point x = +2.00, y = -3.00, z = +1.50. A is a constant.

(b) Find the electric field at a point 7.5 cm from the center of a conducting sphere. The sphere has a charge

of 3.5 × 10-9 C and a diameter of 20.0 cm.

(d) Given a non-conducting sphere of radius 5.00 cm whose total charge (uniformly distributed) is

+9.00 × 10 C. Find the work needed from the external world to move one electron from R = 7.00 cm-9

to R = 11.25 cm. R is measured from the center of the sphere.

18. (a) Calculate the work necessary to assemble the charge

configuration shown.

(b) Find the work needed to bring a charge !2Q from

infinity to point P. P is halfway between the two

charges.

19. Consider a long cylindrical non-conductor. The outside

oradius of the cylinder is R , and the charge distribution inside the cylinder is modeled by D = AR , where A is a 2

oconstant and R # R (artificial, but it keeps the math simple).

(a) Calculate the electric field at any interior point at a distance R from the center of the cylinder.

o(b) Using V(R) = 0 at R = 0, find the value of the potential at a point R where R # R . The sign of V must

be clearly stated for the case where the charge density is positive.

(c) Obtain a formula for A if the cylinder has a linear charge density of 8 C/m. What are the units of A?

20. (a) Calculate the work needed to assemble 3 protons (q = +1.60 × 10 C) into an equilateral triangle-19

1.00 × 10 m on a side. -10

(c) If 15.0 coulomb of charge is uniformly distributed along a wire 2.00 km long, calculate the electric

field at the mid point of the wire and 1.50 cm from the center of the wire and perpendicular to the wire.

(d) Calculate the term in a for the expression given below. a << x. 6

(e) An electric potential is given by V = Axy (1 - z ). Find the y component of the electric field at the3 2

point x = +1.00, y = +2.00, z = +5.00.

21. In the geometry shown the charge +Q is at x = +a and the

charge -2Q

is at x = +2a. The point P is at x = 0, y = +a.

(a) Find the electric potential at point P (with the

standard convention for V = 0).

(b) If Q = 750 pC and a = 1.35 cm, calculate a

numerical value for the potential at P, including

the sign.

(c) How much work (numerical answer) is required

to bring an electron from infinity to point P?

22. (a) If the electric potential in a region of space is calculated as V = Ax y z , calculate the z component of2 2 3

the electric field.

(b) For the function (1 - x) , calculate a numerical result for the coefficient of the term in x (assume-3/2 3

x < 1).

(c) A long rod of non-conductor has a uniform linear charge density given by 8 = +2.75 × 10 C/m. -5

Calculate the magnitude of the electric field 3.20 cm away from the center of the rod (the radius of the

*rod is < 3.20 cm).

(d) The electric field near the surface of a conductor is found to be +372 V/m. Calculate the surface charge

density on the conductor at the point where this electric field is measured.

(e) An electron is accelerated from rest by a potential difference of 10.0 volts. Calculate its velocity.

23. (a) For the configuration of charges shown, calculate the potential at

point P (numerical value) with the standard convention for V = 0.

(b) How much work is required (with sign) to bring an electron from

infinity to point P?

1 2Q = +4.50 × 10 C; Q = -3.27 × 10 C;-6 -6

3Q = +8.50 × 10 C; a = 1.27 cm-6

A24. A long hollow tube of nonconductor has an inside radius of R and an outside radius

Bof R . The nonconductor has a uniform charge density D (C/m ). The charge3

density is zero everywhere else. State in words the shape of the Gaussian surface

that you use in each case.

A(a) Calculate the electric field for R < R .

A B(b) Find the electric field for R < R < R (in the interior of the nonconductor).

B(c) Calculate the electric field for R > R (outside the tube, assuming R is less

than the length of the tube).

25. (a) Determine the work, in Joules, to bring an electron and proton from far apart to a distance of 1.00 ×

10 m apart. !10

(b) Calculate the equivalent capacitance between A and B for the network shown. All capacitances are

3.00 pF.

(c) A 4.75 × 10 F capacitor is charged to 150 volts. Find the energy stored in this capacitor. !6

(d) The electric potential in a region of space is given by V = Ax y z, where A is a constant. Calculate the2 3

magnitude of the z component of the electric field at the point x = 3, y = 2, z = 4. [All quantities in

appropriate SI units.]

(e) A parallel plate capacitor has square plates 0.100 m on a side that are 1.00 × 10 m apart. If this!3

capacitor stores 3.25 x 10 J of energy, find the electric field between the plates. There is no dielectric!4

between the plates.

26. A square distribution of charges is set up as shown. The side of the square is

"a."

(a) Calculate the electric potential, in volts, at point C, the center of the

square.

(b) Calculate the electric potential, in volts, at point D, the middle of the

bottom side of the square.

(c) Find the work, in Joules, necessary to bring a charge Q = 6.00 × 10 C!3

to point C, from infinity.

1 2Q = +1.50 × 10 C Q = !2.50 × 10 C!3 !3

3 4Q = +4.75 × 10 C Q = !3.25 × 10 C!3 !3

a = 1.20 × 10 m!2

27. (a) For the arrangement shown, calculate the equivalent capacitance between A and B.

1(b) If the battery is 150 V, what is the potential across C ?

5(c) If the battery is 150 V, what is the charge on C ?

1 2C = 1.00 pF C = 2.00 pF

3 4C = 3.00 pF C = 4.00 pF

5C = 5.00 pF

28. (a) A capacitor is charged to 225 volts. The value of the capacitance is 600 pF. Calculate the energy

stored in this capacitor.

(b) If the electric potential in region of space is given by v = Bx y z , where B is the constant, calculate the4 2 !2

y component of the electric field (including sign) at the point x = 2, y = !2, z = 4.

(c) A parallel plate capacitor has dimensions of 4.00 cm by 125 cm. The plates are 0.125 mm apart and

there is a dielectric with 6 = 4.25 between the plates. Calculate the capacitance of this capacitor.

(d) If the earth (use R = 6000 km) is found to have a potential of exactly 1.00 million volts, what is the

charge on the earth?

(e) Each of the capacitors in the network shown has the same value

C. Calculate the equivalent capacitance between A and B.

29. A battery of 150 V is connected between points A and B.

(a) Calculate the equivalent capacitance between points A and B.

3(b) Calculate the charge and potential on capacitor C .

1 2 3 4C = 60.0 pF; C = 45.0 pF; C = 35.0 pF; C = 25.0 pF

30. (a) All capacitors have the same value, C. Calculate the equivalent capacitance between a and b.

(b) A 2.50 pF capacitor is charged to 150 volts. Calculate the energy stored in this capacitor in joules.

(c) A positive charge, +Q, and a negative charge, - Q, are a distance

2a apart. Calculate the electric potential half-way between them,

at point P.

(d) If bismuth has 3.00 × 10 charge carriers/cm , calculate the drift17 3

velocity, in m/s, in a square wire 1.00 mm on a side carrying a

current of 4.50 A.

(e) A 1.50 :F capacitor is charged to 200 volts and disconnected from the power source. Now a dielectric

of 6 = 3.00 is inserted fully in the capacitor. Calculate the change in the stored energy in the capacitor

and state whether it is an increase or decrease.

31. For the network shown, the switch S is closed long enough for the capacitors

to be fully charged, and then it is opened.

(a) Calculate the potential and charge for each capacitor.

(b) Now, with the switch still open, a dielectric with 6 = 4.00 is inserted

3into C . Calculate the new values of charge and potential for each

capacitor.

1 2 3C = 1.20 :F; C = 1.50 :F; C = 0.50 :F; g = 2.00 V

32. (a) Given an electric potential of the from V = Ax y z , calculate the magnitude of the electric field at the3 2 5

point x = 2, y = 4, z = 6.

(b) Given the function , calculate the coefficient of the term in x in the4

binomial expansion.

(c) The electric field just above the surface of a sphere of radius 5.00 cm is 1250 N/C. Calculate the total

charge on the sphere.

(d) In moving an electron from point A to point B 4.00 × 10 J of work is done from the outside world. -19

B ACalculate the potential difference V - V , with sign.

(e) If all capacitors have the same value, C, calculate the effective capacitance between

a and b. _____________________________________________________

33. The points shown are all along the x-axis.

(a) Determine the electric field, magnitude and direction, at

point A.

(b) Calculate the electric potential at point A using the usual

assumption about V = 0.

(c) Calculate the electric field, magnitude and direction, at point B.

(d) Find the electric potential at point B.

B34. The capacitor system shown is connected to a battery with V = 180 V.

(a) Calculate the equivalent capacitance between points A and B.

6(b) Calculate the potential across C .

3(c) Calculate the charge on C

1 4C = 1.00 :F C = 4.00 :F

2 5C = 2.00 :F C = 5.00 :F

3 6C = 3.00 :F C = 6.00 :F

035. A very long cylinder of non-conductor and radius R is charged with a

ovolume charge density D given by D = BR between R = 0 and R = R . B is a

constant.

o(a) Calculate the magnitude of the electric field at any arbitrary distance R from the center (for R < R ).

o(b) Calculate the magnitude of the potential difference between R = 0 and R = R .

o(c) If the rod is negatively charged, what is the sign of V(R ) - V(0)?

(d) Calculate the energy stored in the electric field for 1 meter length of this cylinder between R = 0 and

oR = R /2.

236. With the switch S closed, a dielectric of 6 = 3.50 is inserted into C . Then the

2switch is opened. After opening the switch the dielectric is removed from C .

At this point, with the switch still open, calculate:

1(a) the voltage and charge on C ,

2(b) the voltage and charge on C

3(c) the voltage and charge on C , and

(d) the voltage between points A and B.

(Capacitances values given are without dielectric)

1 2 3 BC = 20 pF; C = 10 pF; C = 40 pF; V = 150 V

o37. Consider a sphere of nonconductor whose radius is R and 6 = 1.00. It is positively charged with a charge

density given by D = A/R, where A is a constant and R the distance from the center of the sphere. Calculate the

o oenergy stored in the electric field between R = R /2 and R = R .

1 238. (a) Calculate the equivalent capacitance of the series combination of 3 capacitors: C = 40.0 pF, C = 60.0

3pf, C = 45.0 pf.

(b) A 25.0 pf capacitor is charged to 100.0 volts. The battery is then disconnected. A second capacitor,

uncharged, of capacitance 60.0 pf is connected in parallel with the first. Find the new potential across

the combination.

(c) A parallel plate capacitor has an area of 3.50 m , and a separation of 0.145 mm. The dielectric material2

has a dielectric constant of 4.25. Calculate its capacitance in picofarads. (Be sure to use right units.)

(d) A dielectric material has a dielectric strength of 12,500 V/mm. If it is used in a capacitor at a thickness

of 0.130 mm, find the breakdown voltage for that capacitor.

(e) A potential is given by V = Ax y where A is constant and x and y are the coordinates. Find the x2

component of the electric field.

39. A certain material has a dielectric constant of 3.2 and a dielectric strength of 16 × 10 volts/meter. If it is used6

as the dielectric material in a parallel-plate capacitor, what minimum area may the plates of the capacitor have

in order that the capacitance be 7.0 × 10 :F and that the capacitor be able to withstand a potential difference-2

of 4000 volts?

40. Given that bismuth, a semimetal, has a density of 9.80 g/cm and a carrier density of 7.65 × 10 cm , calculate3 18 -3

the number of carriers per atom. Bismuth has an atomic mass of 209.

41. Aluminum has an atomic weight of 27.0 and a density of 2.699 g/cm . Assume that aluminum has 3 free3

electrons per atom. If an aluminum wire of diameter 1.00 cm carries a current of 1200 amperes, find

(a) the drift velocity, and

(b) the current density.

42. Copper has a density of 8.92 g/cm and an atomic mass of 63.6. Assume that it has one free electron per atom. 3

If a copper wire of cross section 3 × 10 m carries a current of 10.0 amperes.-6 2

(a) Find the drift velocity.

(b) Find the current density.

43. The switch is closed for a long time and then opened. All capacitors are given without dielectric. Now a

2dielectric with 6 = 3.25 is inserted in C .

1 2(a) Calculate the charge and voltage on C and C .

(b) With conditions as in (a) the switch is closed. How much additional

charge flows from the battery to the capacitors?

g = 150 V

1C = 12.5 :F

2C = 6.0 :F

o44. Given a long cylindrical nonconductor of radius R , calculate the total energy stored in the electric field inside

the cylinder per length L. The cylinder has a uniform charge density of D C/m .3

45. In a sample of pure silicon, the carrier density is found to be 1.25 × 10 per atom. Silicon has a density of-6

2.400 × 10 kg/m , and an atomic mass of 28.0. The silicon is in the form of a long, square rod 1.00 m on a3 3

side and carries a current of 3.25 × 10 A along the rod.-3

(a) Calculate the current density.

(b) Find the drift velocity.

46. A wire has a radius is 0.25 mm and a length of 20 cm. The wire is in the center of a metal tube whose inner

radius is 1.55 cm. A potential of 4000 V is applied between the wire and the tube. Calculate the energy stored

in the electric field between the wire and tube.

1 247. In the circuit shown, initially S is closed and S is open. The capacitor

values are given without dielectric. Initially a dielectric with 6 = 2.70 is in

2C .

1 2(a) Now S is opened, and then S is closed. Calculate the potential across

3C .

1 2 2(b) With S still open and S still closed, the dielectric is moved from C

3to C . Calculate the potential and charge on all capacitors.

1g = 175 V C = 100 pF

2 3C = 30 pF C = 150 pF

48. A cylindrical capacitor is created with the outer radius of the inner

A Bconductor R and the inner radius of the outer conductor R .

(a) Calculate the capacitance (numerical value) of a 2.5 m length of this

capacitor.

(b) What is the energy being stored between R = 0.50 cm and R = 1.00

cm in a 2.5 m length of this capacitor (numerical value). The

capacitor has a potential difference of V = 175 V across it.

A BR = 0.25 cm R = 1.25 cm

49. (a) Calculate the coefficient of the x term for the given expression: 4

(b) Calculate V(B) - V(A).

(c) Calculate the equivalent capacitance between a and b if all

capacitors have the value C.

(d) The charge carrier density (n) for copper is 8.47 × 10 e /m . The cross section of the wire is 6.02 cm . 28 - 3 2

If the wire carries 11 Amps, what is the drift velocity?

(e) Calculate the current density in a copper wire with circular cross section carrying a current of 12.5 A.

The diameter of the wire is 2.74 mm.

1 250. Given two conducting spheres of radius R and R . A total

charge Q is placed on sphere 2. Sphere 1 is uncharged. Now

the two are connected with a wire.

(a) Calculate the charge on each sphere.

(b) Calculate the potential of each sphere.

(c) Calculate the electric field at the surface of each sphere.

1 2 1 251. First S is closed with S open. Then S is opened, and finally, S is closed. Calculate the

voltage and charge for each capacitor in the final configuration

g = 12 Volts

1C = 10 :F

2C = 23 :F

3C = 6 :F

52. Given a sphere of nonconductor with a charge density given by

o (for r < R ).

T(a) Find Q .

o(b) Find E when r < R .

o(c) Find E when r > R .

o(d) Find the energy stored in the region o < r < R .

o o(e) If D < 0, what is the sign of V(0) - V(R )? Explain.

53. (a) Calculate the capacitance of a series combination of a 3.75 pF and a 5.75 pF capacitor.

(b) All capacitors have the same value, C. Calculate the equivalent capacitance between points A and B.

(c) Given the function , calculate the term involving x4

using the binomial expansion.

(d) A heavy copper wire carries a current of 11.7 × 10 A. The wire is a square with sides of 0.600 cm. If3

copper has 8.48 × 10 electrons/cm that are free to move, determine the drift velocity in the wire.22 3

(e) Find the energy stored in a parallel plate capacitor at a voltage of 165 V. The plates have an area of

0.175 m and a separation of 3.10 mm. There is no dielectric. 2

54. Given the circuit shown.

(a) Calculate the equivalent capacitance between points A

and B.

1(b) If the battery is 175 V, what is the potential across C ?

5(c) If the battery is 175 V, what is the potential across C ?

3(d) If the battery is 175 V, what is the charge on C ?

1 4C = 5.00 pF C = 2.00 pF

2 5C = 6.00 pF C = 4.00 pF

3 6C = 3.00 pF C = 7.00 pF

Use at least 4 significant figures in intermediatecalculations.

55. In the diagram shown the switch is closed for a long time and then opened.

3Now a dielectric of 6 = 3.25 is inserted into C . Capacitors values are given

without dielectric.

1(a) Find the new voltage across C .

2(b) Find the new voltage across C .

(c) Find the charge that flows and the direction it flows when the

dielectric is inserted.

1 2 3g = 15.0 V; C = 155 pF; C = 125 pF; C = 250 pF

56. Two long conducting cylinders have the same center. The outer radius of the inner

tube is 2.50 mm. and the inner radius of the outer tube is 11.7 mm. A potential of 2.75

V is applied between the inner and outer tubes.

(a) Find the energy per unit length stored between r = 2.50 mm and r = 4.50 mm.

(b) What is the capacitance per meter of this structure?

57. (a) Calculate the equivalent capacitance between A and B if all capacitors have

the same value, C.

(b) Calculate the term involving x using the binomial expansion for the expression . 6

Calculate a number or fraction for the coefficient.

(c) Calculate the current density in a copper wire with a total current of 110 Amp if the wire is circular with

a diameter of 2.25 mm.

(d) An electric potential has the form V = Ax y z , where A is a constant. Calculate the x-component of the4 2 3

electric field.

(e) Two capacitors of values 2.00 :F and 4.00 :F, are in series as shown. The battery

is 155 V. What is the energy stored in the combination of capacitors?

58. Given the circuit shown.

(a) Calculate the equivalent capacitance between points A and B.

6(b) If the battery potential is 315 volts, what is the potential across C ?

4(c) If the battery is 315 volts, what is the potential across C ?

2(d) If the battery is 315 volts, what is the charge on C ?

Note: Keep 4 significant figures in intermediate calculations.

1 4C = 2.50 pF C = 3.00 pF

2 5C = 3.50 pF C = 5.00 pF

3 6C = 2.00 pF C = 7.00 pF

59. Given that bismuth, a semi-metal, has a density of 9.80 g/cm and a charge3

carrier density of 7.65 × 10 /cm . Bismuth has an atomic mass of 209.18 3

(a) Determine the number of charge carriers per atom.

(b) Calculate the drift velocity of the charge carriers in a wire of cross section 1.75 mm carrying a current2

of 9.50 A.

o60. Given a sphere of non-conductor of radius R with a charge density distribution

given by

o ofor r < R , and D = 0 everywhere else. Take D as negative.

T(a) Find the total charge on the sphere, Q .

(b) Calculate the electric field at an arbitrary point r inside the sphere.

(c) Determine the potential difference , including its sign.

61. Note: If you use a sign convention different from that

used in class, be sure to indicate that.

For the circuit shown.

(a) How many total junctions are there?

(b) How many independent junction equations can

be written?

(c) How many branches are there?

(d) How many unknown currents will be needed to

analyze this circuit?

(e) Write the loop equations for loop Î, going

clockwise around the loop.

1 1 2 2R = 1000 S; g = 100 V; R = 900 S; g = 55 V;

3R = 700 S

62. For the circuit shown, use the values given below.

3(a) Calculate the current in R , including sign. The

arrows show the positive direction.

1(b) Find the magnitude of the potential across R .

4(c) Find the current, magnitude and sign, through R . The

arrow shows the positive direction.

1 5R = 400 S R = 800 S

2 1R = 300 S g = 115 V

3 2R = 150 S g = 17.0 V

4 3R = 600 S g = 95.0 V

63. Use the numerical values below for the components in the circuit. At t = 0

the switch is closed after being opened for a long time.

3(a) Calculate the current in R when 0.75 time constants have elapsed

after t = 0.

1(b) Calculate the current in R at t = 4.

(c) Calculate the charge on the capacitor after 0.75 time constants have

elapsed since t = 0.

(d) Using the full loops and junctions, and NO shortcuts, calculate the

time constant for charging the capacitor. (Use the currents indicated.)

1 2 3g = 125 V; R = 3000 S; R = 2500 S; R = 1750 S; C = 2.75 × 10 F-6

64. (a) Calculate the capacitance of the series combination of a 4.0 pf capacitor and a 2.0 pf capacitor.

(b) Calculate the capacitance of the parallel combination of a 3.0 pf capacitor and a 1.5 pf capacitor.

(c) An electron is subjected to an electric field of 370 V/m. Find its acceleration.

(d) A 1.5 volt battery is connected across a 100 pf capacitor. Find the charge on the capacitor.

(e) The electric potential in a region of space is given by V = Cx . Find the electric field.2

65. Calculate the capacitance of two concentric metal spheres, one of radius a and the other of radius b.

1 3 266 When the switch is closed, C and C have no dielectric and C has a dielectric

1 3totally filling the capacitor with K = 4.00. The values of C and C given are

2without dielectric, C is with the dielectric.

(a) Find the potential and charge on each capacitor.

2 1(b) Now the switch is opened and the dielectric is moved from C to C .

Find the new values of the charge and potential for each capacitor.

1V = 65.0 V C = 3.00 pF

2 3C = 16.00 pF C = 8.00 pF

A B A67. In the arrangement shown, S is originally closed and S is open. Then S is opened

Bfollowed by closing S .

B(a) With S closed, find the charge on each capacitor.

B(b) With S still closed, find the voltage across each capacitor.

B A(c) Leaving S closed, close S . How much additional charge flows

Athrough S ?

1 2 3V = 150 V; C = 25 pf; C = 30 pf; C = 50 pf

68. In the diagram shown the switch is closed for a long time and then opened. Now a dielectric of 6 = 4.00 is

3inserted into C .

1(a) Find the new voltage across C .

2(b) Find the new voltage across C .

(c) Find the charge that flows and the direction it flows when the

dielectric is inserted.

1 2 3g = 12.0 V; C = 155 pF; C = 125 pF; C = 250 pF

1 1 269. In the diagram shown g = 150 V , C = 30 pf and C = 40 pf.

(a) The switch S is closed and then a dielectric of dielectric constant 6 =

24.00 is inserted in C . Find the potentials and charges on the capacitor.

2 1(b) Now the switch is opened and the dielectric is moved from C to C .

Find the new potentials and charges in the capacitors.

1 2 1 2If you cannot do part (a) use V , V , Q and Q as starting charges and

potentials for part (b).

70. In the arrangement shown, both switches are initially open and all capacitors are fully discharged. Now close

1 2 1 2 2 1S while leaving S open. Then open S with S still open. Finally, close S with S remaining open. Work

parts (a) and (b) with the switches in this state.

(a) Calculate the charge on each capacitor.

(b) Calculate the voltage across each capacitor.

2 1(c) Leaving S closed with the charges and voltages as in (a) and (b), close S . How much additional

charge flows from the battery to the capacitors?

1 2 3V = 1.25 V; C = 3.00 pF; C = 4.00 pF; C = 2.00 pF

71. In the arrangement shown both switches are initially open and all capacitors are fully discharged. First, close

1 2 1 2 2 1S with S open. Then open S , leaving S open. Finally, close S with S still open. Work part (a) with the

switches in this state. Capacitance values are without dielectric.

(a) Calculate the charge on the potential difference across each capacitor.

2 1 2(b) With S closed and S open, as in (a), insert a dielectric with 6 = 5.00 into C . Recalculate the charge

and potential difference for each capacitor.

1 2 3V = 110 V; C = 7.25 pF; C = 2.00 pF; C = 5.00 pF

o o72. A conducting sphere has a charge Q = 4.55 × 10 C and a radius R , where R = 0.37 m. -6

o oCalculate the energy stored in the electric field between R = 2R and R = 3R .

1 273. Initially all capacitors are uncharged. Then switch S is closed leaving S open.

1 2(a) Calculate the potential and charge for capacitors C and C .

1 2(b) Now S is opened, and then S is closed. Calculate the potential and

charge on each capacitor.

2 1(c) With S still closed, close S and calculate the additional charge that

flows from the battery to the capacitors.

1 2C = 120 :F C = 70 :F

3 BC = 30 :F V = 125 V

74. (b) A wire carries a current of 2500 A. The cross section of the wire is a square, with each edge of the

square having a length of 0.100 cm. If copper has 8.48 × 10 electrons/m free to move, calculate the28 3

drift velocity of the free electrons in the wire.

(c) Calculate the effective resistance between points a and b. All resistors are 5.00 S.

(d) A resistor carrying 1.07 A of current is dissipating 47.2 watts. Calculate its resistance.

(e) Copper has a resistivity at room temperature of 1.7 × 10 ohmAm. Find the resistance at room-8

temperature of 3.00 miles of copper wire whose diameter is 1.25 mm.

75. (a) If each resistor is 2.00 ohms, calculate the effective resistance

between (a) and (b).

(d) A heavy copper wire carries a current of 12500 Amps. The wire is square with each side 0.500 cm. If

copper has 8.48 × 10 electrons/m free to move, calculate the drift velocity in this wire.28 3

(e) A 12,500 ohm resistor is dissipating 0.752 watts of power. Find the current in this resistor.

76. (a) Calculate the effective capacitance between a and b. All capacitors have a value of 350 pF.

(c) Silicon has an atomic mass of 28.0 and a density of 2.40 × 10 kg/m . If a sample of silicon has a3 3

charge carrier density of 3.50 × 10 cm , how many carriers per atom does it have? 1 6 -3

77. (a) A current of 150 A is flowing in an aluminum wire of diameter 2.00 mm. Aluminum has three free

electrons per atom, an atomic mass of 27.0 and a density of 2.7 × 10 kg/m . Find the drift velocity.3 3

(b) Find the effective resistance between a and b. All resistors have the

same value.

(e) Calculate the effective capacitance between a and b if all capacitors have

the value of 175 Pf.

78. (a) Given a simple circuit consisting of a resistance of 15 × 10 ohms and a capacitance of 6.0 pF, find the4

time constant.

(b) Given an RC circuit. The capacitance begins to discharge at t = 0. If the charge on the capacitor is

3.0 × 10 C at t = 0, what is it at t = 2 RC?-9

(c) A current of 1.5 amperes passes through a resistance of 100 ohms. What is the power dissipated?

(d) Bismuth has an atomic mass of 209.0. If it has 3.0 × 10 charge carriers per cm and a density of 9.8017 3

g/cm , how many charge carriers per atom does it have?3

(e) Calculate the effective resistance between points a and b.

79. (a) Given a simple circuit consisting of a resistance of 2.7 × 10 ohms and a capacitance of 15,000 pF, find5

the time constant.

(b) Given an RC circuit. The capacitor is discharged at t = 0. The capacitor is charged through the

resistance R from a battery of 100 V. If the capacitance is 3000 pF, what is the charge on the capacitor

at t = 1.5 RC?

(c) A current of 15 amperes passes through a resistance. The potential across the resistance is 110 volts.

What is the power dissipated in the resistance?

(d) Antimony has an atomic mass of 121.8 and a density of 6.68 g/cm . If antimony has 3.0 × 10 charge3 18

carries per cm , how many carriers per atom does it have?3

(e) Calculate the effective resistance between points a and b.

80. A parallel plate capacitor is constructed with a plate area of 0.40 m and a plate separation of 2 mm. A3

dielectric plate, of dielectric constant 3.5, completely fills the volume between the plates. The capacitor is

connected to a battery of 90 V, and then disconnected. Then the dielectric plate is removed.

(a) What is the charge on the plates?

(b) What is the potential between the plates.

(c) What is the electric field between the plates?

(d) If the capacitor is now reconnected to the battery, how much charge flows, and in which direction?

81. Given the arrangement of capacitors shown with the values shown.

(a) Calculate the effective capacitance between A and B.

(b) If a potential of 150 volts is applied between A and B, find the charge

4on C .

(c) If a potential of 150 volts is applied between A and B, find the

2potential across C .

1 2 3C = 15.00 :F; C = 5.00 :F; C = 10.00 :F

4 5 6C = 12.00 :F; C = 10.00 :F; C = 20.00 :F

82. (a) Calculate the equivalent resistance between a and b for the network given.

All resistors have the same value, R.

(b) A resistor of R = 1250 S is dissipating 3.27 × 10 W. Calculate the current in it.-1

(c) Aluminum has a resistivity at room temperature of 2.83 × 10 ohm@m. Calculate the cross sectional-8

area for an aluminum wire if it is to have a resistance of 1.00 S per mile.

4(d) For the network shown, calculate the potential across R .

1R = 125 S

2R = 150 S

3R = 375 S

4R = 425 Sg = 100 V

83. In the circuit shown the switch is open for a long time and then closed at t = 0.

2(a) At t = 0, calculate the voltage across R .

3(b) Calcualte the voltage across R at t = 4.

(c) After 1.00 time constant has elapsed after t = 0, calculate the charge

on the capacitor.

(d) Calculate a numerical value for the time constant. Show all work with

no short cuts from more advanced classes.

1 2R = 150 S R = 175 S

3R = 115 S g = 100 V

C = 130 :F

84. A wire of radius r = 0.100 mm is at the center of a conducting cylinder of inner radius R = 1.25 cm. There is a

potential difference of 525 V between the wire and the cylinder. Calculate the energy stored in a 2.00 m length

of this structure between r and R/2.

85. (a) Calculate the equivalent capacitance between (a) and (b). All

capacitors are 3.00 pF.

(b) Calculate the effective resistance between (a) and (b). All resistors

have a value of 100 ohms.

(c) Calculate the drift velocity for a current of 12.0 A in a circular cross section copper wire of radius

1.50 mm. Take the charge carrier density as 5.00 × 10 m .29 -3

(d) Calculate the power being dissipated in the 150 ohm resistor.

(e) Antimony has an atomic mass of 121.8 and a density of 6.68 g/cm . If antimony has 3.00 × 10 charge3 18

carriers per cm , how many carriers per atom does it have?3

86. (a) A copper rod, with circular cross section, has a radius of 1.200 cm. If it carries a current of 47,500 A,

calculate the current density.

(b) A 375 S resistor dissipates power at the rate of 4.40 Watts. Calculate the current.

(c) Calculate the equivalent capacitance between points A and B. All C's are

25 pf.

(d) Calculate the equivalent resistance between X and Y.

(e) Calculate the drift velocity for a current of 12,000 A in a copper rod with square cross section 1.00 cm

on a side. Take the charge carrier density as 6.00 × 10 m .29 -3

87. (a) For the circuit shown, calculate the time constant.

R = 1.45 × 10 S3

1C = 25.0 pF

2C = 40.0 pF

(b) Calculate the capacitance of a parallel plate capacitor if the plates are square, with sides of 0.250 m, a

separation of 0.220 mm, and a dielectric of 6 = 4.50 filling the space between the plates.

(c) A copper wire has a diameter of 2.30 mm. If it carries a current of 17.2 A, calculate the magnitude of

the current density in A/m .2

(d) Antimony (a semimetal) has atomic mass of 121.8 and a density of 6.68 g/cm . If we assume it has3

1.25 × 10 charge carriers per atom, what is the density of charge carries per m ?-4 3

(e) All resistances in the network have a resistance R. Calculate the effective

resistance between a and b.

88. For the circuit shown all capacitance values are given WITHOUT

DIELECTRIC. Initially, all capacitors are discharged. A dielectric with 6 =

34.00 is inserted in C , and then the switch is closed. After a time long

enough to equilibrate, the switch is opened. Now the dielectric is moved

3 2from C to C . The switch remains open.

(a) Calculate the charge and potential on each capacitor before the

3 2dielectric is moved from C to C .

(b) Calculate the charge and potential on each capacitor after the

3 2dielectric is moved from C to C .

1g = 150 V C = 32.0 :F

2 3C = 15.0 :F C = 10.0 :F

89. For the circuit shown, and the values given, calculate:

1(a) the potential across R ;

2(b) the potential across R ;

3(c) the power being dissipated in R ;

7(d) the power being dissipated in R .

1 2g = 150 V g = 100 V

1 2R = 650 S R = 350 S

3 4R = 120 S R = 50 S

5 6R = 70 S R = 80 S

7 8R = 120 S R = 80 S

90. (a) If all capacitors have the same value, C, calculate the

effective capacitance between a and b.

(b) Silicon has an atomic mass of 28.0 and a density of 2.33 g/cm . If impurities are added so there is3

1.00 × 10 charge carriers per atom, calculate the number of charge carriers per cm .-5 3

(c) A copper wire carries a current of 157 A. If the diameter of the wire is 2.00 mm, what is the current

density?

(d) If the current in (c) is carried by electrons and the drift velocity is 1.00 × 10 m/s, what the density of-4

charge carriers?

(e) If 60 volts is applied between A and B, calculate the charge on the 15 pF

capacitor.

o o91. A sphere of non-conductor has a volume charge density give by D = A/R , for R < R , where R is the radius of2

the sphere.

A A o(a) Calculate the electric field at a point R = R , where R < R (inside the sphere).

(b) Calculate the magnitude of the electric potential difference between the surface of the sphere and an

Barbitrary point in the interior that is a distance R from the center.

(c) Evaluate the constant A in terms of the total charge on the sphere.

92. (a) Calculate the effective resistance between A and B if all resistors

have the value of 3.00 ohms.

(b) Calculate the effective value of the capacitance between A and B if

all capacitors have the value 30.0 pF.

(c) If g = 75.0 V, what is the power being dissipated in the 25 Sresistor?

(d) Calculate the drift velocity (in m/s) in a silver wire with current =

4.20 A that has a circular cross section with r = 1.20 × 10 m. -3

Silver has 5.90 × 10 electrons/cm that are mobile. 22 -3

(e) A parallel plate capacitor is constricted using a material whose

dielectric constant is 4.27. If it is necessary to have the plate

separator at 1.27 mm, what is the area needed to give C = 17.0 pF?

1 2 193. For the network shown, S is closed for a long time with S open. S

2 1is then opened and S closed. S remains open.

2 1(a) With S closed, find the potential on C .

2 4(b) With S closed, calculate the charge on C .

2 2(c) With S closed, what is the charge on C ?

batt 1V = 120 V C = 250 pF

2 3C = 150 pF C = 100 pF

4C = 225 pF

94. Given the network shown.

(a) How many total junctions are there?

(b) How many unknown currents are there?

(c) Write down the loop equations for interior

loops A and B. Traverse the loops clockwise.

(d) Write down the junction equation for junction

C.

(e) How many loop equations will be needed for a

complete solution?

95. (a) Calculate the equivalent capacitance between a and b. All

capacitors are 7.00 :F.

(b) Find the effective resistance between a and b if all resistors are 9.00

S.

(c) Determine the power being dissipated in the 150 S resistor. V = 125 V.

(d) A given metal has a resistivity of 4.70 × 10 ohm @m. If a wire is to have a resistance of 2.00 S per-8

mile, calculate its cross-sectional area.

(e) The 150 pF capacitor is charged to 100 volts. The battery is removed and then the switch is closed.

Calculate the charge on the 100 pF capacitor.

96. In the network shown, all resistors have a resistance of 5.00 ohm.

(a) Calculate the effective resistance between a and b.

(b) If a 75.0 volt potential is applied between a and b, what is the current in the

resistor labeled x?

(c) With the same 75.0 volt potential between a and b, calculate the potential

across y.

97. Given the network shown. Use the sign convention used in

class, or make clear to the grader what convention you use.

(a) How many total junctions are there?

(b) How many total unknown currents are there?

(c) Write the junction equation for junction C using the

current designations given.

(d) Write the loop equation for interior loop (A) going

clockwise around the loop. Use the current

designations given.

98. For the circuit shown, all capacitance values are given WITHOUT

4DIELECTRIC. Initially there is a dielectric with 6 = 2.70 in C only.

The switch S is closed, and then opened. With the switch open the

4 4dielectric is removed from C . After the dielectric is removed from C

and with the switch still open, find:

1(a) the charge and potential on C ;

2(b) the charge and potential on C ;

4(c) the charge and potential on C .

1 2 3C = 450 :F; C = 220 :F; C = 120 :F;

4 BC = 85 :F; V = 125 V

99. (a) Calculate the equivalent capacitance between a and b if all capacitors have

the same value, C.

(b) Calculate the equivalent resistance

between a and b if all resistors have the

same value, R.

(c) A capacitor of value 16.5 pF is charged to a potential of 250 V. Calculate the energy stored in this

capacitor.

(d) Assume a value for the electric field near the earth's surface of 150 V/m, directed vertically. Calculate

the energy stored in a volume of 2.50 m near the earth's surface. 3

(e) The resistivity of copper is 8.00 × 10 ohmAm. Calculate the resistance of 1500 m of round copper wire-8

of diameter 1.00 mm.

100. For the circuit shown:

(a) How many TOTAL junctions are there?

(b) How many mathematically independent

junction equations can be written?

(c) Write the junction equations for junctions A

and B in the drawing.

(d) How many mathematically independent loop

equations can be written?

(e) Write loop equations for loops I and II in

the drawing. For ease in grading, go

clockwise around the loop. [Tell us if you

use a sign convention different from that

used in class.]

101. The capacitor values above are given without dielectric. The

switch is closed and a dielectric with 6 = 3.00 is inserted in

3 C .

(a) With the switch still closed, calculate the potential and

2charge on C .

(b) Now the switch is opened and remains open. The

3 1dielectric is moved from C to C . Find the new value

2of the potential and charge on C .

(c) Calculate the potential between a and b with the switch

in the same position as in (b).

1g = 120 V C = 12.0 :F

2 3C = 20.0 :F C = 6.00 :F

102. (a) A copper wire has a diameter of 0.950 mm. It carries a current of 4.75 A. Calculate the current density.

(b) Silicon has a density of 2.40 × 10 kg/m and an atomic mass of 28.0. If, in a given sample of silicon3 3

there are 2.25 × 10 carries per atom, calculate the density of charge carriers per m , that is, n. !5 3

(c) Calculate the equivalent resistance between a and b if all

resistors have the same value R.

(d) Calculate the charging time constant for the circuit shown.

R = 7500 S; C = 25.0 :F

(e) Given an RC circuit. The charge on the capacitor is 4.25 × 10 C at t = 0. It begins to discharge at !12

t = 0. What is the charge at t = 1.75 J?

103. Given the network shown. Use the sign convention

used in class, OR make it clear to the grader what sign

convention you use.

(a) How many TOTAL junctions are there?

(b) How many total unknown currents are there?

(c) How many loop equations will be needed?

(d) Write the junction equation for junction (A)

using the current designations given.

(e) Write the loop equation for the interior loop

designated (B). Start at the lower left corner and

go clockwise around the loop.

104. For the circuit shown, the switch is closed at t = 0, after being open for a long time.

(a) Calculate the charge on the capacitor at t = 4.

1(b) Calculate the current in R at t = 2J.

(c) Calculate the time constant (numerical value) for charging

the capacitor.

(d) After the switch has been closed for a long time, it opened at

a new t = 0. Write an expression for the voltage on the

capacitor as a function of time with all quantities evaluated

numerically.

1 2g = 175 V; R = 350 S; R = 475 S; C = 65.0 pF

105. Initially the switch is closed.

(a) Calculate the potential and charge on each capacitor

3(b) Now the switch is opened and dielectric with 6 = 4.00 is inserted in C .

None of the other capacitors have any dielectric. Calculate the

potential and charge on all of the capacitors in this new situation.

1 2 3 4g = 150 V; C = 100 pF; C = 35.0 pF; C = 25.0 pF; C = 65.0 pF

106. Silver has a density of 10.5 g/cm and an atomic mass of 108. Assume that silver has one free charge carrier3

per atom. If a silver wire of diameter 2.300 mm carries a current of 10.7 A, find

(a) the current density;

(b) the number of charge carriers per cubic meter for

silver;

(c) the drift velocity.

107. Given the network shown. Use the sign convention used

in class, OR make it clear to the grader what sign

convention you use.

(a) How many TOTAL junctions (nodes) are there?

(b) How many total unknown currents are there?

(c) How many mathematically independent loop

equations will be needed?

(d) Write the junction equation for junction (A) using

the current designations given.

(e) Write the loop equation for the interior loop

designated (B). Start at the lower left corner and

go clockwise around the loop.

108. For the circuit shown and using the methods discussed in class:

1(a) Calculate the current in R .

2(b) Calculate the potential across R .

5(c) Calculate the current in R .

1 1R = 500 S g = 100 V

2 2R = 700 S g = 50 V

3 3R = 200 S g = 150 V

4R = 100 S

5R = 150 S

109. (a) Each resistor has a value of 3 ohms. Calculate the

equivalent resistance between a and b.

(b) Calculate the equivalent capacitance of three capacitors 3.0, 5.0 and 9.0 :F in series.

(c) A copper wire has a diameter of 1.75 mm. It carries a current of 4.25 A. Calculate the current density.

(d) Calculate the charging time constant if R = 17.5 × 10 S and C has the value3

of 12.8 pF.

(e) A capacitor of 4.75 :F is charged to 150 V. The battery is disconnected and then a dielectric is

inserted. The dielectric is 6 = 4.12. Calculate the potential in the capacitor after the dielectric has been

inserted.

110. Only currents needed for (c) and (d) are shown.

(a) For the network shown, how many mathematically

independent junction (node) equations can be written?

(b) For the network shown, how many mathematically

independent loop equations can be written?

(c) Write the loop equation for the loop A . Use the sign

conventions discussed in class, or make a clear

statement that you are using a different convention.

(d) Write the junction equation at the junction labeled B .

111. For the circuit shown, the capacitor is initially uncharged. The switch is

closed at t = 0.

4(a) Calculate the current in R at t = 0.

3(b) Calculate the current in R at t = 4.

(c) Calculate the charge on the capacitor at t = 4.

(d) Calculate the charge on the capacitor afer 1.5 time constants have

elapsed from t = 0.

(e) Using the methods discussed in class, calculate the time constant

for charging this capacitor.

112. (a) For the circuit given, circle and number number all of the junctions.

How many independent junction equations can be written?

(b) How many loop equations are needed in addition to the junction

equations to completely solve for the unknown currents?

(c) Write down the junction equation for the junction labeled A. Use

the current labels and directions given.

(d) Write down the loop equations for the interior loops labeled (1) and

(2). Start each at A and go clockwise. Signs are important. Use the

current labels and directions given.

113. (a) How many independent junction equations can be written for

the circuit shown?

(b) How many independent loop equations?

(c) Using the current labels and direction given, write a set of

junction and loop equations that can be used to analyze the

circuit. (To simplify grading, go clockwise around your loops.)

114. In the circuit shown, the capacitor is initially uncharged, and the switch S is

closed at time t = 0.

(a) Calculate the charge on the capacitor at t = 4.

(b) Find the voltage across the capacitor when t = 1.50 J.

1(c) Calculate the current in R at t = 0.

(d) Find the time constant of the circuit (numerical value).

Note: No short cuts learned in other courses are permitted.

1 2 3R = 150 kS; R = 175 kS; R = 125 kS; C = 250 pF; g = 55.0 V

115. In the circuit shown, the switch is closed at t = 0. Calculate the numerical values for the following quantities.

(No short cuts such as Thevenin's Theorem, or other material from advanced courses permitted.)

(a) The voltage on the capacitor at t = 4.

(b) The voltage on the capacitor at t = 2J.

1(c) The voltage across R at t = 0.

(d) The time constant for the circuit.

1 2 3g = +150 V; R = 150.0 S; R = 150.0 S; R = 50.0 S;

4R = 50.0 S; C = 3.00 mF

116. In the circuit shown, switch S is connected to point A for a long time.

(a) What is the potential at C?

(b) Now S is switched to point B. What is the potential at C after a long time?

(c) How much charge flows through R from the time the switch is thrown to B

until current stops flowing in R?

(d) Find the current in R as a function of time. Be sure to put numerical values

for all quantities, including the time constant.

1 2 3 4C = 10 pF; C = 20 pF; C = 10 pF; C = 20 pF; R = 10 S; g = 12.0 V5

117. The capacitor is originally uncharged. The switch is placed at A for 0.100 s.

Then it is moved to B. Call this t = 0.

(a) Calculate the charge on the capacitor at t = 0.050 s.

2(b) Calculate the current through R at t = 0.075 s.

(c) At t = 0.075 s, the switch is moved back to A. How much charge is

added to C before equilibrium is attained?

1 2g = 12.0 V; C = 6.75 :; R = 12,500 S; R = 17,200 S

3R = 23,500 S

118. In the circuit shown the capacitor is initially discharged. The switch is placed

in position A for 5.0 seconds, and then moved to position B.

3(a) Calculate the potential across R 7.0 seconds after the switch is moved

to B.

(b) Calculate the charge on the capacitor 7.0 seconds after the switch is

moved to B.

(c) The capacitor is charged for 5.0 seconds with the switch at A. Then

the switch is opened and a dielectric of 6 = 3.00 is inserted in the

capacitor. Then the switch is moved to B. Now calculate the charge and the potential on the capacitor

7.0 seconds after the switch is thrown to B.

1 2 3g = 12.0 V; C = 12.0 × 10 F; R = 6.7 × 10 S; R = 4.2 × 10 S; R = 7.8 × 10 S-6 5 5 5

119. In the circuit shown the switch S is closed at t = 0.

2(a) Calculate the current in R just after the switch is closed.

2(b) Calculate the current in R a very long time after the switch is

closed.

(c) Calculate the voltage across the capacitor a very long time

after S is closed.

(d) Calculate the charge on the capacitor 125 :s after the switch

is closed.

No short cuts learned in other classed are allowed. Part (d) requires the full solution as discussed in class.

1 2 3R = 150 S; R = 250 S; R = 250 S; g = 125 V; C = 1.65 × 10 F-6

120. Given the circuit shown.

(a) How many (total) junctions are there?

(b) How many unknown branch currents are needed to completely

characterize the circuit.

2 7 3(c) Set R = 0 and R = 0, and calculate the current in R .

1 4 2 5g = 150 V; R = 45 S; g = 85 V; R = 240 S;

1 6 2 7R = 95 S; R = 55 S; R = 75 S; R = 150 S;

3 8R = 225 S; R = 180 S

121. Given the circuit shown. Currents are NOT numbered in

sequence.

(a) How many total junctions are there? Circle the junctions and

label A, B, C, D, etc.

(b) Write the proper number of junction equations to give

independent equations. Use the current labels and directions

from the drawing.

(c) How many independent loop equations are needed for a

complete solution? Show how you arrive at this number.

(d) Write loop equations for the interior loops labeled I and II.

Go clockwise around the loop to simplify grading.

122. For the circuit shown and values given, calculate:

4(a) the magnitude of the potential difference across R ;

3(b) the current through R , magnitude and sign, with the direction

of the arrow taken as positive;

1(c) the power being dissipated in R .

1 1 2 2g = 10.0 V; R = 175 S; = 25.0 V; R = 200 S

3 3 4g = 37.0 V; R = 350 S; R = 475 S

123. For the circuit shown and using the current arrows given,

calculate:

1(a) the current I , magnitude and sign;

3(b) the power being dissipated in R ;

5(c) the power being dissipated in R .

1 2 1g = 1.0 V; g = 35.0 V; R = 100 ohms

2 3 4R =150 ohm; R = 200 ohms; R = 75 ohms

5R = 55 ohms

124. (a) For the circuit shown the switch is closed after being open for a long time.

1i. Calculate the potential across R at t = 0.

1ii. Calculate the potential across R at t = 4.

iii. Calculate the charge on the capacitor after 1.70 time constants have

elapsed from t = 0.

(b) For this part the switch opened at t = 0 after being closed for a long time.

2Calculate the current in R after 1.27 × 10 s have elapsed.-3

(c) From the beginning, using loops and juntions as discussed in class, and no

shortcuts, calculate the time constant for charging the capacitor.

1 2 3g = 135 V; R = 275 ohms; R = 125 ohms; R = 95 ohms; C = 4.5 × 10 F-6

125. Use the numerical values below for the components in the circuit shown. At

t = 0 the switch is closed after being open for a long time.

3(a) Calculate the current in R after 1.25 time constants have elapsed after t = 0.

2(b) Find the current in R at t = 4.

(c) What is the charge on the capacitor at t = 4?

(d) Calculate, using loops and junctions as discussed in class, the time constant

for charging the capacitor.

2g = 375 V; R = 2,500 S; C = 1.50 × 10 F; -6

3 1R = 750 S; R = 1,500 S

126. (a) As drawn, with no simplification, how many

TOTAL junctions are there in this circuit?

(b) How many independent junction equations can

be written?

(c) How many mathematically independent loop

equations are needed for a complete solution of

this circuit?

(d) Write the junction equation for (A), using the

current labels given.

(e) Write the loop equations for loop (L), going

around the loop in a clockwise sense.

3127. (a) Calculate the current in R .

1(b) Calculate the power being dissipated in R .

2(c) Calculate the voltage in R .

1 2 3g = 150 volts; R = 67.0 S; R = 200 S; R = 100 S

128. For the circuit shown the switch is open for a long time and then closed at t = 0.

The capacitor is initially uncharged.

2(a) Calculate the current in R at t = 0.

3(b) Calculate the current in R at t = 0.

3(c) Calculate the current in R at t = 4.

(d) Calculate the charge on the capacitor after 1.7 time constants have

elapsed after t = 0.

(e) Using full loops and junctions, and no short cuts learned in other

classes, calculate the time constant for charging the capacitor.

(f) After C is charged, the switch is opened. Calculate the time constant for

discharge of the capacitor.

1 2 3g = 175 V; R = 100 S; R = 125 S; R = 150 S; C = 300 :F

129. For the circuit shown:

(a) How many TOTAL junctions are there?

(b) How many TOTAL branches are there?

(c) How many independent LOOP equations can

be written?

(d) Write the junction equation for junction (A).

(e) Write the LOOP equation for loop (B) going

clockwise around the loop. If you use a sign

convention different from that used in class,

make that clear.

130. Given the circuit shown.

1(a) Calculate the current in R .

2(b) Calculate the current in R .

3(c) Calculate the current in R .

6(d) Find the power being dissipated in R .

1 3 2 4g = 125 V; R = 1100 S; g = 150 V; R = 1400 S;

1 5 2 6R = 700 S; R = 400 S; R = 900 S; R = 200 S

131. In the circuit shown, the switch is closed at t = 0, after being open for a

long time.

(a) Calculate the charge on the capacitor at t = 0.

3(b) Find the current in R at t = 4.

1(c) Find the current in R at t = 0.

2(d) Calculate the current in R after 2.00 time constants have elapsed.

(e) Using full loops and junctions, as shown in class, calculate the time

constant for charging the capacitor.

1 2 3g = 175 V; R = 125 S; C = 3.25 × 10 F; R = 140 S; R = 170 S-6

132. (a) Calculate the z component of the electric field of the potential given by .

(b) If the resistivity of copper is 1.7 × 10 S-m, calculate the resistance of 2.00 kilometers of copper wire-8

whose diameter is 0.300 mm.

(c) All resistors have the same value R. Calculate the effective

resistance between a and b.

(d) If all capacitors have the same value, C, calculate the effective

capacitance between a and b.

(e) Calculate the drift velocity for an electron in a gold wire carrying 2.00 A of current. The wire is circular

in cross section and has a diameter of 2.00 mm. Gold has an atomic mass of 197 and density of 19.3

g/cm . Assume gold has 1 electron per atom that is free. 3

133. A 125 V battery is connected between points a and b in the drawing shown.

6(a) Calculate the potential and charge on C .

5(b) Calculate the potential and charge on C .

3(c) Calculate the potential and charge on C .

1 4C = 3.00 pF C = 6.00 pF

2 5C = 4.00 pF C = 12.00 pF

3 6C = 5.00 pF C = 2.00 pF

134. Given the circuit shown.

(a) How many junctions are there?

(b) How many independent junction equations can be written?

(c) How many unknown currents are there?

(d) Using the current designations given, write the junction

equations for junction (A).

(e) Using the current designations given, write the loop equation

going all the way around the outside and going clockwise. If

you use a sign convention different from that used in class,

give a clear explanation.

135. After being open for a long time, switch S is closed at t = 0, and

remains closed.

(a) Calculate the charge on the capacitor after 1.50 time constants

have elapsed.

(b) Calculate the time constant for charging the capacitor. Show

all work clearly as was done in class. Obtain a numerical

value for J.

(c) After the switch S has been closed for along time, it is opened.

Calculate the numerical value of the time constant for

discharging the capacitor.

1 2 3R = 200 S; R = 400 S; R = 300 S; C = 175 pF

Treat V symbolically wherever needed.

136. (a) Assume a car battery has an emf of 12.0 volts and an internal resistance of 1.20 × 10 S. If it delivers-2

125 A, what is the power dissipated in the internal resistance? ______________________________

(b) For the same battery as in (a), what is the power delivered to the load (careful, a hasty answer may be

wrong.) _______________________________

(c) If all resistors have the same value R, what is the

effective resistance between a and b.

________________________________

(d) If all capacitors have the value C, what is the

effective capacitance between a and b?

_________________________________

(e) The capacitor is charged to 12.0 V and the battery is removed. The switch S is closed at t = 0. What is

the charge on the capacitor at t = 2.00 s? _____________________________________________

R = 1.44 × 10 S C = 2.57 :F6

137. Calculate the drift velocity for a wire carrying a current of 2.50 A. The wire is made of bismuth. It has a cross

section area of 1.24 × 10 m . Assume Bi has the following properties: Atomic mass = 209.0; !6 2

density = 9.78 g/cm ; number of charge carries per atom: 1.00 × 10 . (Yes, 10 . It is not a misprint.)3 !4 !4

138. Given the circuit shown.

1(a) Calculate the magnitude of the current in R .

2(b) Calculate the magnitude of the current in R .

3(c) Calculate the power being dissipated in R .

(d) Determine the direction of the conventional

2current in R (to the right or to the left). Clearly

state your reasoning.

1 2g = 12.00 V R = 25 S

2 3g = 3.50 V R = 100 S

1 4R = 200 S R = 150 S

139. For the circuit shown, the switch is closed at t = 0 after being

open for a long time.

3(a) At t = 4, what is the magnitude of the current in R ?

3(b) At t = 0, what is the magnitude of the current in R ?

(c) What is the charge on the capacitor after 1.80 time

constants have elapsed after t = 0.

(d) Calculate the time constant for charging the capacitor

using full loops and junctions as done in class. No

shortcuts from more advanced classes.

1g = 120 V R = 750 S

2 3R = 200 S R = 425 SC = 4.00 :F

140. Calculate the capacitance per meter of two long parallel wires a distance a apart. (Note: The total electric field

cannot be calculated from Gauss' Law.)

141. For the circuit shown: (use the method shown in class)

1(a) Find the current in R .

2(b) Determine the potential across R .

4(c) Calculate the current in R .

1 2g = 15.0 V g = 12.0 V

1 2R = 500 S R = 400 S

3 4R = 300 S R = 200 S

5R = 150 S

1 2142. Initially S is closed and S is open.

1 2 3(a) Calculate the effective capacitance for C , C , C .

3(b) Calculate the electric charge and potential on C .

2(c) Calculate the electric charge and potential on C .

1 2Now S is opened and then S is closed.

4(d) Calculate the charge and potential on C .

1 2 3 4 g = 150 V; C = 2.0 :F; C = 4.0 :F; C = 5.0 :F; C

= 2.5 :F

143. (a) If the switch is open for a long time and then closed at t = 0, calculate

the charge on C two time constants after t = 0.

2(b) For the same circumstances as above, calculate the current in R two

time constants after the switch is closed.

(c) Using the circuit techniques we have studied and not techniques from

more advanced courses, calculate the time constant for charging the

capacitor (numerical value). Show ALL work.

1 2 3g = 175 V; R = 250 S; R = 300 S; R = 475 S; C = 130 pF

144. In the circuit shown the switch is closed at t = 0 after being open for a long

time.

1 2 3(a) At t = 0, calculate the potentials across R , R and R .

1 2 3(b) At t = 4, calculate the potentials across R , R and R .

(c) At t = 4, calculate the charge on the capacitor.

(d) Calculate the time constant (numerical answer) for charging and for

discharging the capacitor.

1 2 3R = 1.20 × 10 S; R = 2.75 × 10 S; R = 0.85 × 10 S4 4 4

C = 350 pF; g = 150 V

145. Given the circuit shown and using the current arrows given, calculate

1(a) the current I , magnitude and sign;

5(b) the power being dissipated in R ;

3(c) the potential (magnitude, no sign) across R ;

3(d) the sign of I . Show all work or no credit will be given.

1 1g = 15.0 V R = 75 S

2 2g = 175.0 V R = 155 S

3 3g = 25.0 V R = 97 S

4R = 38 S

5R = 49 S

146. In the circuit shown the switch is left open for a long time and then

closed at t = 0.

(a) What the potential across the capacitor at t = 4?

3(b) Find the potential across R at t = 2J (2 time constants).

(c) Use loops and junctions as shown in class to calculate the time

constant for charging the capacitor. No shortcuts learned from

other classes may be used. You may, of course, simplify the circuit

by combining resistors if possible.

(d) The switch is closed for a long time and then opened. Determine the time constant for discharge. (Full

loops and junctions calculations are NOT needed.)

147. For the circuit shown, the capacitor is initially uncharged. The switch is closed at t = 0.

2(a) Find the magnitude of the current in R at t = 0.

1(b) Calculate the potential across R at t = 4.

(c) Determine the charge on the capacitor after 1.50 time constants have

elapsed after t = 0.

(d) Show explicitly, as discussed in class, how to calculate the time

constant for this system. Show clearly how you define the currents.

A BShort cuts discussed in other classes are not permittedI = +7.75 AI = !6.25 A

148. For the circuit shown the switch is open for a long time and then closed at t

= 0.

(a) Calculate the charge on the capacitor at a very long time after t = 0.

1(b) Find the current in R at t = 0.

2(c) Find the current in R when 1.20 time constants have elapsed after t

= 0.

(d) Determine the time constant for charging the capacitor. ashow your

work in detail.

1 2 3R = 1500 S; R = 800 S; R = 2200 S; C = 1.32 × 10 F; g = 200 V!6

1149. For the circuit shown the capacitor is initially uncharged. Initially S is

2 1closed. At t = 0 S is closed and S is still closed.

(a) Calculate the charge on the capacitor at t = 4,

(b) Find the voltage across the capacitor when t = 1.50 J.

1(c) Calculate the current in R at t = 0.

1(d) With the capacitor fully charged S is opened. Calculate the time

constant for discharging C.

1 2(e) With S and S closed, calculate (numerical value) the time constant for

charging C. (No short cuts from advanced courses allowed.)

1 2 3g = 100 V; R = 150 kS; R = 175 kS; R = 125 kS; C = 800 pF

150. For the circuit shown, the switch is closed at t = 0, after being open for a long time.

(a) Calculate the charge on the capacitor at t = 4.

(b) Calculate the charge on the capacitor after 0.75 time constants have

elapsed

1(c) Calculate the current in R after 1.20 time constants have elapsed.

(d) Using full loops and junctions as shown in class, calculate a numerical

value for the time constant for charging the capacitor.

1 2 3g = 375 V; R = 150 S; R = 175 S; R = 235 S; C = 1.55 :F

Data: Use these constants (where it states, for example, 1 ft, the 1 is exact for significantfigure purposes).

1 ft = 12 in (exact)

1 m = 3.28 ft

1 mile = 5280 ft (exact)

1 hour = 3600 sec = 60 min (exact)

1 day = 24 hr (exact)

earthg = 9.80 m/s = 32.2 ft/s2 2

moong = 1.67 m/s = 5.48 ft/s2 2

1 year = 365.25 days

1 kg = 0.0685 slug

1 N = 0.225 pound

1 horsepower = 550 ftApounds/s (exact)

earthM = 5.98 × 10 kg24

earthR = 6.38 × 10 km3

sunM = 1.99 × 10 kg30

sunR = 6.96 × 10 m8

moonM = 7.35 × 10 kg22

moonR = 1.74 × 10 km3

G = 6.67 × 10 NAm /kg-11 2 2

k = 9.00 × 10 NAm /C9 2 2

og = 8.85 × 10 F/m-12

electron chargee = -1.60 × 10 C-19

electronm = 9.11 × 10 kg-31

N = 6.02 × 10 atoms/gmAmole23

N = 6.02 × 10 atoms/kgAmole26