Electricity and MagnetismReview 1: Units 1-6

Mechanics Review 2 , Slide 1

Review Formulas

Electric PotentialPotential energy per

unit charge

b

a

baba ldE

q

UV

Electric Potential Scalar Function that can be

used to determine E VE

Gauss’ LawFlux through closed

surface is always proportional to charge

enclosed

0

encQAdE Gauss’ Law

Can be used to determine E field

SpheresCylinders

Infinite Planes

Electric FieldForce per unit charge

Electric FieldProperty of Space

Created by ChargesSuperposition q

FE

Coulomb’s LawForce law between

point charges q2

q12,122,1

212,1 r̂

r

qkqF

ConductorsCharges free to move

SpheresCylinders

Infinite Planes Gauss’LawField Lines &

Equipotentials ABCD

ABCD

ABCD

ABCD

Fiel

d Lin

es

Equipotentials

Work Done By E Field

b

a

b

a

ba ldEqldFW

b

ababa ldEqWU

Change in Potential Energy

Applications for Conductors

What Determines How They Move?

They move until E = 0 !

E = 0 in conductor determines charge

densities on surfaces

Example

Consider two point charges q1 and q2 located as shown. 1. Find the resultant electric field due to q1 and q2 at the location of q3.2. Find the resultant force on q3. 3. Find the electric potential due to q1 and q2 at the location of q3.

Use components

Example: Spherical Symmetry

A solid insulating sphere of radius R has uniform charge density ρ and carries total charge Q. Find the Electric field everywhere.

Rx

y

Q

r00

encenc QEA

QAdE

Choose a suitable Gaussian surface: A sphere

Calculate the chargeenclosed within the Gaussian surface for r > R and for r < R

encenc VQ

24 rA

2204 r

kQ

r

QE

r

R

kQE

3For r > R: For r < R:

3)3/4( R

Q

V

Q

Example

A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q.Find the Potential difference between two points inside the sphere A and B at distances rA and rB .

R

x

y

Q

rdrErVrVVB

A

r

rAB )()()(

rR

kQrE

3)(

From the previous problem we know that for r < R:

)(2

)()( 223 ABAB rr

R

kQrVrVV

Example: Cylindrical Symmetry

Find the electric field at a distance r from a line of positive charge of infinite length and constant linear charge density λ.

Choose suitable Gaussian surface: A cylinder

Calculate the chargeenclosed within the Gaussian surface

02 r

E

Example: Planar Symmetry

Find the electric field at a distance r due to an infinite plane of positive charge with uniform surface charge density σ.

Choose suitable Gaussian surface: A cylinder perpendicular to the plane

Calculate the chargeenclosed within the Gaussian surface 02

E

Example

Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.a) What is E everywhere?

neutral conductor

r1

r2

y

x+3Q

Use Gaussian surface = sphere centered on origin

0

encQAdE

r < r1

24 rEAdE

20

3

4

1

r

QE

r > r2

20

3

4

1

r

QE

r1 < r < r2

0E

Example

Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.a) What is E everywhere? b) What is charge distribution at r1?

neutral conductor

r1

r2

y

x+3Q

0

encQAdE

r1 < r < r2

0Er1

r2

+3Q 0encQ

21

1 4

3

r

Q

22

2 4

3

r

Q

Example

Suppose we give the conductor a charge of -Qa) What is E everywhere?b) What are charge distributions at r1 and r2?

r1

r2

-3Q

+++

+

+

+

++

+

+ ++

+

+

+

+++

+2Q

+3Q0E

r1 < r < r2

r < r1

20

3

4

1

r

QE

r > r2

20

2

4

1

r

QE

0

encQAdE

Example

Charge q1 = 2μC is located at the origin. Charge q2 = - 6μC is located at (0, 3) m. Charge q3 = 3.00 μC is located at (4, 0) m  Find the total energy required to bring these charges to

these locations starting from infinity.

23

23

13

31

12

21

r

qqk

r

qqk

r

qqkU

Example

Point charge q at center of concentric conducting spherical shell of radii a1, and a2. The shell carries charge Q.What is V as a function of r?

1. Spherical symmetry: Use Gauss’ Law to calculate E everywhere

2. Integrate E to get V metal

+Qa1

a2

+q

cross-section

Charges q and Q will create an E field throughout space

dErVr

r

0

)(

Example

r > a2:

metal

+Qa1

a2

r

+q

cross-section

204

1)(

r

qQrE

0enclosedQ

AdE

Gauss’ law:

a1 < r < a2 :

0)( rE

r < a2 204

1)(

r

qrE

metal

+Qa3

a4

+q

cross-section

Example

To find V:1) Choose r0 such that V(r0) = 0, usual: r0 =

∞

2)Integrate!r > a2 : r

qQrV

04

1)(

a1 < r < a2 : 04

1)( 2

20

aV

a

qQrV

dErVr

r

0

)(

r < a1 : )(0)()( 12 raVaVrV

1204

1)(

a

q

r

q

a

qQrV

Example

A rod of length ℓ has a total charge Q uniformly distributed.Find V at point P.

r

dqkV

2/122 )( ax

dx

r

dq

2/1220

1

axdxkV

l

Example

What is ?

222 )( hxa

dx

r

dq

Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)?

y P

xa

h

x

r

dq = l dx

rr

dqkE ˆ

2

Example

Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)?

rr

dqkE ˆ

2

xa

P

x

r

q1 q

q

dq = l dx

h

xdE

dE

y

222 )( hxa

dxk

r

kdqdE

cosdEdEx

cosdEdEE xx

22)(cos

hxa

xa

Example

Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)?

rr

dqkE ˆ

2

xa

P

x

r

q1 q

q

dq = l dx

h

xdE

dE

y

222 )( hxa

dxk

r

kdqdE

cosdEdEE xx

22)(cos

hxa

xa

2/3220 )(

)(hxa

xadxkPE

a

x

221

ah

h

h

k

Example

The spheres have the same known mass m and charge q and are in equilibrium.

Given the angle θ and the length L, find the charge q on each sphere.

Is equilibrium possible if the charges are different?

Use x and y components

Yes the force will be the same if the product of the charges is the same

Example

A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q.Find the flux through the outside sphere (r ≥ a). What if r < R?

R

x

y

Q

r

0enclosed

SQ

eargchenclosedenclosed VQ