Thermal Physics - Heat Capacity

For an ideal diatomic gas in thermal equilibrium, the ratio of the molar heat capacity at constant volume at very high temperatures to that at very low temperatures is equal to

A. 1
B. 5/3
C. 2
D. 7/3
E. 3

(GR9677 #79)

Solution:

C_V  for diatomic gas: C_V  = C_Vtrans + C_Vrot  + C_Vvib

For very low T, only translational component contributes
→ C_V(T≪) = ³/₂ Nk

For very high T, all 3 components (translational, vibrational, rotational) contribute
→ C_V(T≫) = (³/₂ + 1 + 1) R = ⁷/₂ Nk

Ratio C_V(T≪) / C_V(T≫)  = 7/3

Answer: D

Optics - Dispersion Curve

The dispersion curve shown above relates the angular frequency to the wave number k. For waves with wave numbers lying in the range k₁ < k< k₂ which of the following is true of the phase velocity and the group velocity?

A. They are in opposite directions.
B. They are in the same direction and the phase velocity is larger.
C. They are in the same direction and the group velocity is larger.
D. The phase velocity is infinite and the group velocity is finite.
E. They are the same in direction and magnitude.

(GR9277 #79)

Solution:

Phase velocity: v_p = ω/k
Group velocity: v_g = dω/dk

In the range k₁ < k< k₂:
v_p → positive quantity
v_g → negative quantity (negative slope)
→ v_p and v_g are in opposite directions.

Answer: A

Electromagnetism - Maxwell's Equation

One of Maxwell’s equation is ∇ ∙ B = 0. Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

(GR8677 #79)

Solution:

Gauss' law of magnetism, ∇ ∙ B = 0:
- Total magnetic flux through the gaussian surface is zero
- No magnetic monopole
- B is a constant

(A) and (B) show B is a constant.
Net flux is zero (total lines in = out)
TRUE.

(C) shows magnetic field between two different poles.
Net flux is zero (total lines in = out)
TRUE

(D) shows a magnetic monopole, and
Net flux is not zero (0 lines in, 8 out), and
B is not a constant (spread out).
FALSE

(E) shows magnetic field around a straight wire.
Net flux is zero (no line in and out)
TRUE.

Answer: D

Special Relativity - Rest Mass

A particle leaving a cyclotron has a total relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass of this particle? 

A. 0.25 GeV/c²
B. 1.20 GeV/c²
C. 2.00 GeV/c²
D. 6.00 GeV/c²
E. 16.0 GeV/c²

(GR0177 #79)

Solution:

E² = p²c² + m₀²c⁴

Given: E = 10 GeV and p = 8 GeV/c

100 = (64/c²)c² + m₀²c⁴
m₀²c⁴ = 100 − 64 = 36
m₀ = 6 GeV/c²

Answer: D