Electromagnetism - Maxwell's Equation

Listed below are Maxwell’s equations of electromagnetism. If magnetic monopoles exist, which of the these equations would be INCORRECT?

I. Curl H = J + ∂D/∂t
II. Curl E = −∂B/∂t
III. div D = ρ
IV. div B = 0

A. IV only
B. I and II
C. I and III
D. II and IV
E. III and IV

(GR9277 #13)

Solution:

Gauss' law of magnetism: ∇ · B = 0
→ There is no magnetic monopole
If there is one, ∇ · B ≠ 0

Faraday's law of induction: ∇ × E = −∂B/∂t
→ if there is magnetic monopole, ∇ × E = −∂B/∂t + (current of magnetic monopole)

Answer: D

Electromagnetism - Maxwell's Equation

One of Maxwell’s equation is ∇ ∙ B = 0. Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

(GR8677 #79)

Solution:

Gauss' law of magnetism, ∇ ∙ B = 0:
- Total magnetic flux through the gaussian surface is zero
- No magnetic monopole
- B is a constant

(A) and (B) show B is a constant.
Net flux is zero (total lines in = out)
TRUE.

(C) shows magnetic field between two different poles.
Net flux is zero (total lines in = out)
TRUE

(D) shows a magnetic monopole, and
Net flux is not zero (0 lines in, 8 out), and
B is not a constant (spread out).
FALSE

(E) shows magnetic field around a straight wire.
Net flux is zero (no line in and out)
TRUE.

Answer: D

Electromagnetism - Maxwell Equation

Maxwell’s equations can be written in the form shown below. If magnetic charge exists and if it is conserved, which of these equations will have to be changed

A. I only

B. II only

C. III only

D. I and IV

E. II and III

(GR0177 #41)

Solution:

Gauss Law for magnetism:

• It implies there is no magnetic monopoles.
• If there is magnetic monopole:

Faraday Law of Induction:

• The electric field is equal to the negative of the rate of change of the magnetic flux.
• If there is magnetic monopole, an extra term (involving the current of magnetic monopoles) must be added to the right-hand side of the equation in order to make it self-consistent.

Answer: E