Physics Problems & Solutions: #88

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Electromagnetism - Magnetic Field

A coaxial cable has the cross section shown in the figure above. The shaded region is insulated. The regions in which r

a and b

c are conducting. A uniform dc current density of total current I flows along the inner part of the cable (r

a) and returns along the outer part of the cable (b

c) in the direction shown. The radial dependence of the magnitude of the magnetic field, H, is shown by which of the following?

(GR9677 #88)

Solution:

For r = 0 → H = 0
→ D and E are FALSE

For r = c → H = 0 since I_inand I_out cancel each other
→ A and C are FALSE

Answer: B

Electromagnetism - Capacitor

A parallel-plate capacitor is connected to a battery. V₀ is the potential difference between the plates, Q₀ is the charge on the positive plate, E₀ the magnitude of the electric field, and D₀ the magnitude of the displacement vector. The original vacuum between the plates is filled with a dielectric and then the battery is disconnected. If the corresponding electrical parameters for the final state of the capacitor are denoted by a subscript f, which of the following is true?

A. V_f> V₀
B. V_f< V₀
C. Q_f = Q₀
D. E_f> E₀
E. D_f> D₀

(GR9277 #88)

Solution:

(A), (B) FALSE
The battery is disconnected after the plates is filled with a dielectric
V_f= V₀ = constant

(C) FALSE
Charge: Q = CV

V_f= V₀ and when dielectric is inserted: C_f= κC₀
Q_f = C_f V_f = κC₀V₀ = κQ₀

(D) FALSE
Potential: V = Ed
V_f= V₀ and d is not changed
E_f= E₀

(E) TRUE
Displacement vector in empty space: D₀ = ε₀E
with dielectric constant: D_f = κε₀E
D_f> D₀

Answer: E

Thermal Physics - Maxwell-Boltzmann Statistics

Consider a system of N non-interacting particles confined in a volume V at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let P_F be the pressure exerted by the particles if they are fermions. P_B be the pressure if they are bosons and, and P_C be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

(GR8677 #88)

Solution:

Ideal gas: $PV=NkT \rightarrow P=\frac{N}{V}kT$

Classical (Maxwell-Boltzmann Statistics): $E=\frac{3}{2}kT$

$P=\frac{N}{V}kT\rightarrow P_C = \frac{2}{3} \frac{N}{V} \frac{3}{2} kT=\frac{2}{3} \frac{NE}{V}$

For Bosons (photon):

$P=\frac{N}{V}kT\rightarrow P_B = \frac{1}{3} \frac{N}{V} 3 kT=\frac{1}{3} \frac{NE}{V}$

→

For Fermion: $E=\frac{3}{5}E_F$
E_F

= Fermi Energy, $E_F \gg kT$

For the same temperature:

Answer: B

Electromagnetism - Biot Savart's Law

A segment of wire is bent into an arc of radius R and subtended angle θ, as shown in the figure. Point P is at the center of the circular segment. The wire carries current I. What is the magnitude of the magnetic field at P?

A. 0
B. ^μ₀I^θ/₍₂_π₎_²_R

C. ^μ₀I^θ/₄_πR
D. ^μ₀I^θ/₄_πR_²
E. ^μ₀I/₂_θ_R_²

(GR0177 #88)

Solution:

Biot Savart’s Law:

$\begin{aligned} d \vec{B}&=\frac{\mu_0}{4\pi} \frac{Id \vec{l}\times \hat{r}}{r^2} \\\\\hat{r}&=\vec{r}/r \\d \vec{B}&=\frac{\mu_0}{4\pi} \frac{Id \vec{l}\times \vec{r}}{r^3} \end{aligned}$

$\begin{aligned} &Id \vec{l}\perp \vec{R} \rightarrow Id \vec{l}\times\vec{R}=IR dl \\ &d \vec{B}=\frac{\mu_0}{4\pi} \frac{IR dl}{R^3} \\&dl=Rd\theta\rightarrow d \vec{B}=\frac{\mu_0}{4\pi} \frac{IR^2d\theta}{R^3} \\&B=\frac{\mu_0 I\theta}{4\pi R} \end{aligned}$

Answer: C