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  r  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  r  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_in and I_out cancel each other
→ A and C are FALSE

Answer:  B

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 - Conductor

A conducting cavity is driven as an electromagnetic resonator. If perfect conductivity is assumed, the transverse and normal field components must obey which of the following conditions at the inner cavity walls?

A. E_n = 0, B_n = 0
B. E_n = 0, B_t = 0
C. E_t = 0, B_t = 0
D. E_t = 0, B_n = 0
E. None of the above

(GR9277 #34)

Solution:

Boundary Condition on the surface of perfect conductor:

• E_tangential = 0
• E_normal ≠ 0 → (A) and (B) FALSE
• B_tangential ≠ 0 → (C) FALSE
• B_normal = 0

Answer: D

Electromagnetism - Conductor

A plane-polarized electromagnetic wave is incident normally on a flat, perfectly conducting surface. Upon reflection at the surface, which of the following is true? 

A. Both the electric vector and magnetic vector are reversed.
B. Neither the electric vector nor the magnetic vector is reversed.
C. The electric vector is reversed; the magnetic vector is not.
D. The magnetic vector is reversed; the electric vector is not.
E.  The directions of the electric and magnetic vectors are interchanged.

(GR9277 #36)

Solution:

Perfect conductor: 

Its charge and current are distributed on the surface so that external fields can not penetrate it.

Boundary Condition (BC) on the surface of perfect conductor:

• E_tangential = 0
• E_normal ≠ 0 
• B_tangential ≠ 0
• B_normal = 0 

EM wave: E ⊥ B (perpendicular to each other).

If E normal to the surface of conductor, B is tangential.
BC: E_normal ≠ 0 and B_tangential ≠ 0
Only E is reversed.
B is not reserved, it's parallel to the surface.

Answer: C

Electromagnetism - Conductor

An electromagnetic plane wave, propagating in vacuum, has an electric field given by E = E₀ cos (kx−ωt) and is normally incident on a perfect conductor at x = 0, as shown in figure. Immediately to the left of the conductor, the total electric field E and the total magnetic field B are given by which of the following?

	E	B
A.	0	0
B.	2E0 cos ωt	0
C.	0	2(E0/c) cos ωt
D.	2E0 cos ωt	2(E0/c) cos ωt
E.	2E0 cos ωt	2(E0/c) sin ωt

(GR0177 #61)

Solution:

Perfect conductor: 

Its charge and current are distributed on the surface so that external fields can not penetrate it.

Boundary Condition (BC) on the surface of perfect conductor:

• E_tangential = 0
• E_normal ≠ 0 
• B_tangential ≠ 0
• B_normal = 0 

EM wave: E ⊥ B (perpendicular to each other).

If E normal to the surface of conductor, B is tangential.

BC: E_normal ≠ 0

E_initial is reflected by the conductor with the same magnitude but opposite direction.

Total E = E_initial − E_reflected = 0

(B), (D), (E) FALSE

BC: B_tangential ≠ 0

B is not reflected, it's parallel to the surface.

Total B ≠ 0

(A) FALSE

Answer: C