Electromagnetism - Faraday's Law

A uniform and constant magnetic field B is directed perpendicularly into the plane of the page everywhere within a rectangular region as shown above. A wire circuit in the shape of a semicircle is uniformly rotated counterclockwise in the plane of the page about an axis A. The axis A is perpendicular to the page at the edge of the field and directed through the center of the straight-line portion of the circuit. Which of the following graphs best approximates the emf ε induced in the circuit as a function of time t?

(GR9277 #57)

Solution:

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

Given:
N = 1,
B constant
A increases and decreases uniformly since the wire circuit rotates uniformly
→  the rate of change of A, dA/dt = constant.

ɛ = − BdA/dt  = constant

Only graph (A) shows constant ε.
ε changes periodically from positive to negative since only half of area covered in magnetic field.

Answer: A

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