PHYSICS : THIRD PAPER, February Program 2011 3

Question 1 (2+4+2+2 marks):

Figure 1 shows a network of resistors, and emfs. The emfs have negligible internal

resistance. Also labeled are assumed currents I and i.

(a) Write down two Kirchoff loop equations for the circuit shown.

(b) Solve the above equations to find the values of the currents I and i.

(c) Find the power lost in the ammeter.

(d) Redraw and label the circuit showing the physical magnitudes and directions of the

currents through all the resistors.

A

4 V

6 V

2 Ω

3 Ω

1 Ω

2 Ω I

i

Figure 1: Question 1

PHYSICS : THIRD PAPER, February Program 2011 4

a

3a/2

2a

A

B

C

θP

Figure 2: Question 2

Question 2 (4+2+4 marks):

Point charges of -Q, +2Q and +Q are located at points A, B and C respectively as

illustrated in Figure 2.

(a) Determine the potential V at point P due to all three charges.

(b) Draw a diagram showing the directions of the individual electric field vectors produced

at point P by each of the charges.

(c) Use Coulomb’s law to find an expression for the total electric vector E at point P

due to the three charges, in terms of the parameters given and ijk unit vectors.

PHYSICS : THIRD PAPER, February Program 2011 5

X X X X X

X X X X X

X X X X X

X X X X X

r

B

LA

C

B

Dv t = 0

X

Y

Figure 3: Question 3

Question 3 (2+5+3 marks):

Figure 3 shows two parallel conducting rails, AB and CD, whose ends, A and C, are

connected by a resistance, r. The length of the rails is L. A uniform magnetic field of

magnitude, Bo, exists between the rails, which is perpendicularly downward to the plane

of the rails. A bar, XY , starting at ends, B and D, at time t = 0, slides along these rails

at a constant velocity, v, towards ends the other end. The bar is always perpendicular to

the rails. Dimensions of the system are labeled on the figure.

(a) Draw a diagram of the circuit to indicate the direction of the induced current i that

flows in the bar XY .

(b) Use Faraday’s law to derive an expression for the magnitude of this current, i, in the

bar, in terms of time, t, and the parameters given.

(c) Derive an expression for the force F that acts on the bar XY to keep it moving at

constant velocity v along the rails.

END OF EXAM

1

FOUNDATION STUDIES

EXAMINATIONS

December 2012

PHYSICS

Final Paper

February

Time allowed 3 hours for writing10 minutes for reading

This paper consists of 6 questions printed on 13 pages.PLEASE CHECK BEFORE COMMENCING.

Candidates should submit answers to ALL QUESTIONS.

Marks on this paper total 120 Marks, and count as 45% of the subject.

Start each question at the top of a new page.

2

INFORMATION

a · b = ab cos θ

a× b = ab sin θ c =

∣∣∣∣∣∣i j kax ay azbx by bz

∣∣∣∣∣∣v ≡ dr

dta ≡ dv

dtv =

∫a dt r =

∫v dt

v = u+ at a = −gjx = ut+ 1

2at2 v = u− gtj

v2 = u2 + 2ax r = ut− 12gt2j

s = rθ v = rω a = ω2r = v2

r

p ≡ mv

N1 : if∑

F = 0 then δp = 0N2 :

∑F = ma

N3 : FAB = −FBA

W = mg Fr = µR

g= acceleration due to gravity = 10 m s−2

τ ≡ r× F∑Fx = 0

∑Fy = 0

∑τP = 0

W ≡∫ r2r1

F dr W = F · s

KE = 12mv2 PE = mgh

P ≡ dWdt

= F · v

F = kx PE = 12kx2

dvve

= −dmm

vf − vi = ve ln(mi

mf)

F = |ve dmdt |

F = k q1q2r2

k = 14πε0≈ 9× 109 Nm2C−2

ε0 = 8.854× 10−12 N−1m−2C 2

E ≡limδq→0

(δFδq

)E = k q

r2r

V ≡ Wq

E = −dVdx

V = k qr

Φ =∮

E · dA =∑q

ε0

C ≡ qV

C = Aεd

E = 12q2

C= 1

2qV = 1

2CV 2

C = C1 + C21C

= 1C1

+ 1C2

R = R1 +R21R

= 1R1

+ 1R2

V = IR V = E − IR

P = V I = V 2

R= I2R

K1 :∑In = 0

K2 :∑

(IR′s) =∑

(EMF ′s)

F = q v ×B dF = i dl×B

F = i l×B τ = niA×B

v = EB

r = mq

EBB0

r = mvqB

T = 2πmBq

KEmax = R2B2q2

2m

dB = µ0

4πidl×rr2∮

B · ds = µ0

∑I µ0 = 4π×10−7 NA−2

φ =∫area

B · dA φ = B ·A

ε = −N dφdt

ε = NABω sin(ωt)

f = 1T

k ≡ 2πλ

ω ≡ 2πf v = fλ

y = f(x∓ vt)y = a sin k(x− vt) = a sin(kx− ωt)

= a sin 2π(xλ− t

T)

P = 12µvω2a2 v =

√Fµ

s = sm sin(kx− ωt)

∆p = ∆pm cos(kx− ωt)

3

I = 12ρvω2s2

m

n(db′s) ≡ 10 log I1I2

= 10 log II0

where I0 = 10−12 W m−2

fr = fs

(v±vr

v∓vs

)where v ≡ speed of sound = 340 m s−1

y = y1 + y2

y = [2a sin(kx)] cos(ωt)

N : x = m(λ2) AN : x = (m+ 1

2)(λ

2)

(m = 0, 1, 2, 3, 4, ....)

y = [2a cos(ω1−ω2

2)t] sin(ω1+ω2

2)t

fB = |f1 − f2|

y = [2a cos(k∆2

)] sin(kx− ωt+ k∆2

)

∆ = d sin θ

Max : ∆ = mλ Min : ∆ = (m+ 12)λ

I = I0 cos2(k∆2

)

E = hf c = fλ

KEmax = eV0 = hf − φ

L ≡ r× p = r×mv

L = rmv = n( h2π

)

δE = hf = Ei − Ef

rn = n2( h2

4π2mke2) = n2a0

En = −ke2

2a0( 1n2 ) = −13.6

n2 eV

1λ

= ke2

2a0( 1n2

f− 1

n2i) = RH( 1

n2f− 1

n2i)

(a0 = Bohr radius = 0.0529 nm)

(RH = 1.09737× 107m−1)

(n = 1, 2, 3....) (k ≡ 14πε0

)

E2 = p2c2 + (m0c2)2

E = m0c2 E = pc

λ = hp

(p = m0v (nonrelativistic))

∆x∆px ≥ hπ

∆E∆t ≥ hπ

dNdt

= −λN N = N0 e−λt

R ≡ |dNdt| T 1

2= ln 2

λ= 0.693

λ

MATH:

ax2 + bx+ c = 0 → x = −b±√b2−4ac

2a

y dy/dx∫ydx

xn nx(n−1) 1n+1

xn+1

ekx kekx 1kekx

sin(kx) k cos(kx) − 1k

cos kxcos(kx) −k sin(kx) 1

ksin kx

where k = constant

Sphere: A = 4πr2 V = 43πr3

CONSTANTS:

1u = 1.660× 10−27 kg = 931.50 MeV1eV = 1.602× 10−19 Jc = 3.00× 108ms−1

h = 6.626× 10−34 Jse ≡ electron charge = 1.602× 10−19 C

particle mass(u) mass(kg)

e 5.485 799 031× 10−4 9.109 390× 10−31

p 1.007 276 470 1.672 623× 10−27

n 1.008 664 904 1.674 928× 10−27

PHYSICS: Final Paper. February 2012 8

5Ω

2Ω

1Ω 3Ω

4Ω

battery

ε = 10 V r = 0.5 Ω

I

iA B

Figure 5:

Question 3 ( (2 + 2 + 2 + 2 + 2) + (4 + 6) = 20 marks):

Part (a):

Figure 5 shows a DC network of resistors and a battery with internal resistance r.

(i) Find the total load (external) resistance of the circuit.

(ii) What is the current I drawn from the battery?

(iii) Find the potential drop VAB across the resistor of 2 Ω?

(iv) What is the current i that will flow through the resistor of 2 Ω?

(v) Find the power dissipated from the 5Ω resistor.

PHYSICS: Final Paper. February 2012 9

a

a

b

d

P

Figure 6:

Part (b):

Figure 6 shows a conducting plate of dimensions a × a × b, where a >> b. It carries a

net charge of 2Q.

(i) Draw a sketch diagram showing the electric field lines inside AND outside the

plate. Show clearly also the distribution of the charges in the plate.

(ii) Using Gauss’ law, find the magnitude E of the electric field vector, at a point P

which lies in the middle region of the plate, at a distance d (d << a) from the surface

of the plate. Clearly describe the Gaussian surface that you use. Be careful to give full

and careful justifications, for each and every step of your reasoning, as marks will be

allocated for these reasons.

PHYSICS: Final Paper. February 2012 10

Rr

B

Figure 7:

Question 4 ( (3 + 5 + 2) + (1 + 1 + 1 + 1 + 3 + 3) = 20 marks):

Part (a):

A circular wire with a time-varying radius r and a total resistance of R is immersed in a

time-varying magnetic field of B :

B = Bo + βt2

r = 2t

where β is a positive constant and t >0. The set-up is as shown in Figure 7.

(i) Use Lenz’ law to find the direction of the induced current for t >0. Remember to

explain fully your answer.

(ii) Use Faraday’s law to find an expression for the magnitude of the induced emf E

in terms of Bo, β and t for t >0.

(iii) Hence, find an expression for the magnitude of the induced current for t >0.

PHYSICS: Final Paper. February 2012 11

M

generator

Figure 8:

Part (b):

A frequency generator is used to generate a continuous transverse wave to travel down a

tautly stretched wire of length l = 1400 cm and total mass of m = 35 g, as illustrated in

Figure 8. The wire is under a tension of T . The wave equation in SI units is given as :

y = 0.08 sinπ(0.3x+ 50 t)

For this wave, find

(i) its amplitude A.

(ii) its frequency f .

(iii) its wavelength λ.

(iv) the wave velocity v.

(v) the power P that the source of the wave will need to transmit to the string to

maintain the wave.

(vi) the tension T of the wire in order that the wire can sustain the propagation of

the wave.

PHYSICS: Final Paper. February 2012 12

Question 5 ( (5 + 5) + (3 + 2 + 3 + 2) = 20 marks):

Part (a):

A photoelectric cell has a magnesium electrode. The work function for magnesium

is 3.68 eV. It is found that when the cell is illuminated with light of wavelength

λ=241.24 nm, the maximum velocity of the ejected electrons v=7.2×105 m/s.

(i) Calculate the threshold wavelength for this photocell.

(ii) What must be the frequency of the illuminated light if the ejected electrons are

to have a maximum velocity of 1.7×106 m/s?

Part (b):

The Balmer series of spectral lines for atomic hydrogen, are formed by electron transi-

tions terminating on the n = 2 energy level. Use the Bohr theory for the hydrogen atom

to answer the following questions.

(i) Calculate the two longest wavelengths, and the shortest wavelength, in this

spectral series.

(ii) What is the total energy of an electron in the n = 2 level?

(iii) What is the potential energy of the electron in that orbit?

(iv) Calculate the kinetic energy for that electron.

PHYSICS: Final Paper. February 2012 13

Question 6 ( (2 + 4 + 4) + (5 + 5) = 20 marks):

Part (a):

Plutonium-241 (Pu-241) is a fissile isotope of plutonium, formed when plutonium-240

captures a neutron. Pu-241 undergoes β− decay to Americium-241, which is nonfissile.

An amount of Pu-241 was left in an un-used reactor. After t years, it is found that 10%

of the nuclei has decayed away. After another 67 years, there is only 53% of the nuclei left.

(i) Write an expression for the number of nuclei left N at time t.

(ii) Find the half-life of Pu-241.

(iii) Determine the activity of the Pu-241 at the time t.

Part (b):

Curium was first produced by Glenn Seaborg, Ralph James and Albert Ghiorso,

working at the University of California, Berkeley, in 1944. They bombarded atoms of

Plutonium-239 (atomic number 94) with alpha particles that had been accelerated in a

device called a cyclotron. This produced curium-242 (atomic number of 96).

(i) Write down the full nuclear equation for this reaction. Name any particles emitted.

(ii) Calculate the energy (in MeV) released in a single such reaction, given the masses

below. What form does this energy take?

Pu− 239 239.052163381 uCm− 242 242.058835824 u

α 4.002602 u

END OF EXAM