Special Relativity - Electric Field

The infinity xy-plane is a nonconducting surface, with surface charge density σ, as measured by an observer at rest on the surface. A second observer moves with velocity v relative to the surface, at height h above it. Which of the following expressions gives the electric measured by this second observer?

A.

B.

C.

D.

E. 

(GR9677 #49)

Solution:

E_z = γE_0z 
with γ = 1/(1 − ^v2/_c²)^½ 

E_z = E₀/(1 − ^v2/_c²)^½

For infinity plane: E = σ/2ɛ₀ (show)

For infinity xy-plane: E₀  = σ / 2ɛ₀

E_z =  σ / 2ɛ₀(1 − ^v2/_c²)^½

Answer: C

Nuclear & Particle Physics - Muon

Two horizontal scintillation counters are located near the Earth’s surface. One is 3.0 meters directly above the other. Of the following, which is the largest scintillator resolving time that can be used to distinguish downward-going relativistic muons from upward-going relativistic muons using the relative time of the scintillator signals?

A. 1 picosecond
B. 1 nanosecond
C. 1 microsecond
D. 1 millisecond
E. 1 second

(GR9277 #49)

Solution:

Muons can travel with maximum speed near the speed of light:
v = 3 × 10⁸ m/s

Given s = 3 m,

t = s/v = 3/(3 × 10⁸) = 10⁻⁸

Micro = 10⁻⁶
Nano = 10⁻⁹
Pico = 10⁻¹²

10⁻⁸ second is close to 1 nanosecond.

Answer: B

Quantum Mechanics - Schrodinger Equation

The Hamiltonian operator in the Schrodinger equation can be formed from the classical Hamiltonian by substituting

A. Wavelength and frequency for momentum and energy
B. A differential operator for momentum
C. Transition probability for potential energy
D. Sums over discrete eigenvalues for integrals over continuous variables
E. Gaussian distributions of observables for exact values

(GR8677 #49)

Solution:

Schrodinger Equation: Hψ(x) = Eψ(x)

Hamiltonian:

Momentum operator : 

Answer: B

Thermal Physics - Maxwell-Boltzmann Statistics

In a Maxwell-Boltzmann system with two states of energies ε and 2ε respectively, and a degeneracy of 2 for each state, the partition function is

A. e ^{−ε / kT}
B. 2e^{−2ε / kT}
C. 2e^{−3ε / kT}
D. e^{−ε / kT} + e^{−2ε / kT}
E. 2(e^{−ε / kT}+ e^{−2ε / kT})

(GR0177 #49)

Solution:

Canonical partition function:  Z = Σ_i e^−βE_i
β = 1/kT

For Maxwell-Boltzmann System:  Z = Σ_i g_i e^−βE_i
g_i = degeneracy

Thus, the partition function is
Z = 2e^−βε + 2e^−2βε = 2(e^{−ε / kT} + e^{−2ε / kT})

Answer: E