Electromagnetism - Angular Momentum

Two small pith balls, each carrying a charge q, are attached to the ends of a light rod of length d, which is suspended from the ceiling by a thin torsion-free fiber, as shown in the figure. There is a uniform magnetic field B, pointing straight down, in the cylindrical region of radius R around the fiber. The system is initially at rest. If the magnetic field is turned off, which of the following describes what happens to the system?

1. It rotates with angular momentum qBR².
2. It rotates with angular momentum ¼ qBd².
3. It rotates with angular momentum ½ qBRd.
4. It does not rotate because to do so would violate conservation of angular momentum.
5. It does not move because magnetic forces do no work.

(GR9677 #87)

Solution:

B and C are FALSE.
Angular momentum is finite, it cannot go to infinite.
If d → ∞ , angular momentum → ∞

D. FALSE
Since there is external torque, angular momentum is not conserved.

E. FALSE
The system will rotate due to B.

Answer: A:

Calculation:
Faraday's Law:


with Φ = BπR² and dl = 2πd/2 = πd

 


Torque:

Since there is a force contribution from each charge and by the right-hand-rule their cross products with the moment-arm point in the same direction,



Torque and Angular momentum:

Classical Mechanics - Circular Motion

A particle of mass M moves in a circular orbit of radius r around a fixed point under the influence of an attractive force F = ^K⁄_r³, where K is a constant. If the potential energy of the particle is zero at an infinite distance from the force center, the total energy of the particle in the circular orbit is

A. − ^K⁄_r² 
B. − ^K⁄_2r²
C. 0
D. ^K⁄_2r²
E. ^K⁄_r²

(GR9277 #87)

Solution:

Attractive force = Centripetal Force
 ^K⁄_r³ = ^mv²⁄_r
mv² = ^K⁄_r²

Kinetic energy:  T = ½mv² = ^K⁄_2r²
Potential energy: V(r) = − ∫ F dr = −K ∫ ¹⁄_r³ dr = ^K⁄_2r²
Attractive force → negative potential energy, V(r) = − ^K⁄_2r²
Total energy:  T + V = ^K⁄_2r²  −  ^K⁄_2r² = 0

Answer: C

Thermal Physics - Diatomic Molecule

In a gas of N diatomic molecules, two possible models for a classical description of a diatomic molecule are:



Which of the following statements about this gas is true?

A. Model I has a specific heat c_v = ³⁄₂Nk
B. Model II has a smaller specific heat than Model I
C. Model I is always correct
D. Model II is always correct
E. The choice between Models I and II depends on the temperature

(GR8677 #87)

Solution:

Equipartition of Energy: E = ½ fNkT
Specific heat: c_v = ∂E/∂t = ½ fNk

f = Degree of Freedom (DoF)

(A) FALSE
Model I is diatomic molecules in low temperature
→ 5 DoF (3 translational + 2 rotational)
→ c_v = ⁵⁄₂Nk

(B) FALSE
Model II is diatomic molecules in high temperature
→ 7 DoF (3 translational + 2 rotational + 2 vibrational)
→ c_v = ⁷⁄₂Nk

(C) and (D) are vague answers.

Answer: E

Electromagnetism - Electric Force

Two spherical, nonconducting, and very thin shells of uniformly distributed positive charge Q and radius d are located a distance 10d from each other. A positive point charge q is placed inside one of the shells at a distance d/2 from the center, on the line connecting the centers of the two shells, as shown in figure. What is the net force on the charge q? 

A.     to the left

B.     to the right

C.     to the left

D.     to the right

E.     to the left

(GR0177 #87)

Solution:

Inside thin shell, no charge → E = 0

Outside thin shell,  E = ^kQ/_r² where k = ¹/_4πɛ0

Thus, we just have to consider the force exerted by the opposite sphere, to the left.

r = 10d − ^d/₂ = ^19d/₂

E = ^kQ/_(19d/2)² = ^4kQ/_361d² = ^Q/_361πɛ0d²

F = qE = ^qQ/_361πɛ0d²

Answer: A