Electromagnetism - Polarization

Questions 54-55 concern a plane electromagnetic wave that is a superposition of two independent orthogonal plane waves and can be written as the real part of 

E = x̂ E₁ exp [i(kz −ωt)] +  ŷ E₂ exp [i(kz − ωt + π)]

where k, ω, E₁ and E₂ are real

If the plane wave is split and recombined on a screen after the two portions, which are polarized in the x- and y- directions, have traveled an optical path difference of 2π/k, the observed average intensity will be proportional to

A. E₁² + E₂² 
B. E₁² − E₂² 
C. (E₁+ E₂)² 
D. (E₁− E₂)² 
E. 0

(GR9677 #55)

Solution:

E = x̂ E₁ e^{i(kz − ωt)}  +  ŷ E₂  e^{i(kz − ωt +π )} 

Path difference of 2π/k
E = x̂ E₁ e^{i(kz − ωt)}  +  ŷ E₂  e^{i[k(z + 2π/k) − ωt + π)} 

E = x̂ E₁ e^{i(kz − ωt)}  +  ŷ E₂  e^{i[kz − ωt)} · e^i3π 
e^i3π = −1

E = x̂ E₁ e^{i(kz − ωt)}  −  ŷ E₂  e^{i(kz − ωt)} 

Intensity in the x- directions, I_x = |E₁|²  
Intensity in the x- directions, I_y = |−E₂|² 

I_total = I_x + I_y = E₁² + E₂²

Answer: A

Electromagnetism - Ampere's Law

Questions 54-55

A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current I. The distance between the wire and the left side of the loop is r. The loop is pulled to the right as indicated. 

What is the magnitude of the net force on the loop when the induced current is i?

A. 

B. 

C. 

D. 

E. 

(GR9277 #55)

Solution:

Magnetic force on an electric current: 

For the loop, induced current I = i is parallel to B, and  l = b

 → F = ibB
Ampere's Law for a magnetic field around a straight wire: 
→  

Only choice D fits the above equation.

Answer: D


Complete calculation:

On the left, r  → r
On the right, r  → r + a






Note: since force on the left is directed to the left → negative sign

The magnitude is 

Thermal Physics - Conduction Electrons

The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy kT. Which of the following can best be used to explain this fact?

A. The energy-time uncertainty relation
B. The Pauli exclusion principle
C. The degeneracy of the energy levels
D. The Born approximation
E. The wave-particle duality

(GR8677 #55)

Solution:

The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy kT due to Pauli’s exclusion principle.

See problem GR0177 #76

Answer: B

Classical Mechanics - Conservation of Momentum

A particle of mass m is moving along the x-axis with speed v when it collides with a particle of mass 2m initially at rest. After the collision, the first particle has come to rest, and the second particle has split into two equal-mass pieces that move at equal angles θ  0 with the x-axis, as shown in the figure. Which of the following statements correctly describes the speeds of the two pieces?

A. Each piece moves with speed v
B. One of the pieces moves with speed v, the other moves with speed less than v.
C. Each piece moves with speed v/2
D. One of the pieces moves with speed v/2, the other moves with speed greater than v/2.
E. Each piece moves with speed greater than v/2

(GR0177 #55)

Solution:

Conservation of momentum:



Answer: E