Electromagnetism - Electric and Magnetic Force

A positively charged particle is moving in the xy-plane in a region where there is a non-zero uniform electric magnetic field B in the +z –direction and a non-zero uniform electric field in the +y-direction. Which of the following is a posible trajectory for the particle?

(GR9677 #86)

Solution:

v is in the xy-plane
E is in +y-direction


→ F is in +y-direction
→ particle will be deflected by E in +y-direction

B is in +z-direction


→ F,v, B orthogonal to each other
→ E and B orthogonal to each other

Particle moving in an orthogonal direction with B will exhibit cyclotron (helix shaped motion).                    
Answer:  B

Electromagnetism - Oscilloscope

The circuit shown above is used to measure the size of the capacitance C. The y-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across R, and the x-coordinate of the spot is swept at a constant speed s. The switch is closed and then opened. One can then calculate C from the shape and the size of the curve on the screen plus a knowledge of which of the following?

A. V₀ and R
B. s and R
C. s and V₀
D. R and R'
E. The sensitivity of the oscilloscope

(GR9277 #86)

Solution:

The voltage of a capacitor follows an exponential decay:

V(t) = V₀e^−t/RC 

C depends on V(t), V₀, t, and R.

V₀ is given and known from the beginning of measurement.

The y-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across R. Thus, V(t) is the shape and the size of the curve on the screen.
 
The x-coordinate of the spot is swept at a constant speed s. From this we can calculate time t.

Therefore, we can calculate C from the shape and the size of the curve plus a knowledge of s and R.

Answer: B

Electromagnetism - Faraday’s law

A coil of 15 turns, each of radius 1 centimeter, is rotating at a constant angular velocity ω = 300 radians per second in an uniform magnetic field of 0.5 Tesla, as shown in figure. Assume at time t = 0 that the normal n̂ to the coil plane is along the y-direction and that the self-inductance of the coil can be neglected. If the coil resistance is 9 ohms, what will be the magnitude of the induced current in milliamperes?

A. 225π sin ωt
B. 250π sin ωt
C. 0.08π cos ωt
D. 1.7π cos ωt
E. 25π cos ωt

(GR0177 #86)

Solution:

Ohm’s Law: ɛ = V = IR
→  I = ɛ / R

Faraday's Law: ɛ = − dΦ_B / dt

Magnetic Flux:  dΦ_B =  NB_⊥dA

At time t = 0, the normal to the coil plane is along the y-direction.

It means: n̂ ∥ ŷ → B ∥ A → Φ_B = 0

→ Φ_B =  NBA sin ωt, since Φ_B (t = 0) = 0

Φ_B = − NB₀ πr² sin ωt
ɛ = − dΦ_B / dt = NB₀ πr² ω cos ωt

I =  ɛ / R = (NB₀ωr²/ R) π cos ωt

N  = 15 turns
B₀ = 0.5 Tesla
ω = 300 rad/s
r  = 1 cm =  10⁻²  m
R = 9 ohms

I = (15 ×  0.5 × 300 × 10⁻⁴ / 9) π cos ωt
= (25 × 10⁻³) π sin ωt Ampere
= 25 π sin ωt milliAmpere

Answer: E