Electromagnetism - Electric Potential

Question 3-4: refer to a thin, nonconducting ring of radius R, as shown below, which has a charge Q uniformly spread out on it.

  

The electric potential at a point P, which is located on the axis of symmetry a distance x from the center of the ring, is given by

A. Q / (4πɛ₀x)
B. Q / [4πɛ₀(R² + x²)^1/2]
C. Qx / [4πɛ₀(R² + x²)]
D. Qx / [4πɛ₀(R² + x²)^3/2]
E. QR / [4πɛ₀(R² + x²)]

(GR9677 #03)

Solution:

Electric Potential, V = kQ/r 
with k = 1/4πɛ₀

The distance r of P from the charged ring is r² = R² + x²

V = Q / [4πɛ₀(R² + x²)^1/2]

Answer: B

Electromagnetism - Faraday’s law

A wire is being wound around a rotating wooden cylinder of radius R. One end of the wire is connected to the axis of the cylinder, as shown in the figure. The cylinder is placed in a uniform magnetic field of magnitude B parallel to its axis and rotates at N revolutions per second. What is the potential difference between the open ends of the wire?

A. 0
B. 2πNBR
C. πNBR²
D. BR²/N
E. πNBR³

(GR9677 #47)

Solution:

Faraday's Law: ɛ = − dΦ/dt
with  Φ = NBA  

Given:
N revolution per second
B uniform (constant)
A constant

ɛ = − BA dN/dt
= − BπR²N

|ɛ| = πNBR²

Answer: C

Electromagnetism - Electric Potential

Two large conducting plates form a wedge of angle α as shown-in the diagram above. The plates are insulated from each other; one has a potential V₀ and the other is grounded. Assuming that the plates are large enough so that the potential difference between them is independent of the cylindrical coordinates z and ρ, the potential anywhere between the plates as s function of the angle φ is

A. V₀/α
B. V₀φ/α
C. V₀α/φ
D. V₀φ²/α
E. V₀α/φ²

(GR9277 #12)

Solution:

Boundary conditions:
V(φ = 0) = 0 
V(φ = α) = V₀ 

(A) FALSE
V₀/α does not depend on φ 

(B) TRUE
V₀φ/α = 0 for φ = 0
V₀φ/α = V₀ for φ = α

(C) FALSE. V₀α/φ = ∞ for φ = 0

(D) FALSE
For φ = α, V₀φ²/α  = V₀α 

(E) FALSE
For φ = α, V₀φ²/α  = V₀α

Answer: B

Electromagnetism - Electric Circuit

The battery in the diagram above is to be charged by the generator G. The generator has a terminal voltage of 120 volts when the charging current is 10 Amperes. The battery has an emf of 100 volts and an internal resistance of 1 Ohm. In order to charging current, the resistance R should be set at 

A. 0.1 Ω
B. 0.5 Ω
C. 1.0 Ω
D. 5.0 Ω
E. 10.0 Ω

(GR8677 # 24)

Solution:

Potential difference across the resistor R:

V = IR_t = V_Generator − V_Battery = 120 − 100 = 20 V

Total resistance:

R_t = R + R_internal = R + 1

V = IR_t = I(R + 1) = 10(R + 1) = 20 V
R = 1 Ω

Answer: C

Electromagnetism - Gauss' Law

A cube has a constant electric potential V on its surface. If there are no charges inside the cube, the potential at the center of the cube is 

A. zero
B. V/8
C. V/6
D. V/2
E. V

(GR8677 #52)

Solution:

Gauss’ Law:  

No charges inside the cube →q_enc = 0 → E = 0

Electric field: E = ∇V
E = 0 → V = constant

Since the potential function has to remain continuous everywhere, at the center of the cube, potential is V.

Answer: E

Electromagnetism - RL Circuit

 

In the circuit shown above, the switch S is closed at t = 0. Which of the following best represents the voltage across the inductor, as seen on an oscilloscope?

(GR0177 #40)

Solution:

The voltage drops across the resistor and the inductor → (A), (B), (C) are FALSE.

The time constant when the voltage decreases slowly is



Answer: D

Electromagnetism - RL Circuit

In the circuit shown above, R₂ = 3R₁ and the battery of emf ξ has negligible internal resistance. The resistance of the diode when it allows current to pass through it is also negligible. At time t = 0, the switch S is closed and the currents and voltage are allowed to reach their asymptotic values. Then at time t₁, the switch is opened. Which of the following curves most nearly represents the potential at point A as a function of time t?

(GR8677 #94)

Solution:

When the switch is closed, at time t = 0, the voltage drops through the circuit. Thus, C, D and E are FALSE.

The time constant in the first process:


The time constant, τ  is the time it takes the voltage across the component to either decrease or increase.

When the switch is opened, at time t₁, the voltage increases with the time constant:

Thus,

It means, it takes less time for the voltage to increase than decrease.

Answer: B

Electromagnetism - Electric Potential

The long thin cylindrical glass rod shown above has length l and is insulated from its surroundings. The rod has an excess charge Q uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If k is the constant in Coulomb’s law, the electric potential at a point P along the axis of the rod and a distance l from one end is kQ/l multiplied by 

A. 4/9
B. 1/2
C. 2/3
D. ln 2
E. 1

(GR8677 #98)

Solution:

Line Charge Density:
, and for uniform line charge:

Electric Potential:



Answer: D