00 Chapter 3 Active-Recall Diagnostic Quiz

00 Chapter 3 Active-Recall Diagnostic Quiz (Magnetostatics)

Overview: Test your conceptual understanding and mathematical recall of magnetostatic postulates, Biot-Savart law applications, Ampere’s law, vector magnetic potential A, bound current densities, magnetic boundary conditions, and the Hall effect.

Question 1: Magnetostatic Fundamental Postulates

State the differential and integral forms of the two fundamental postulates of magnetostatics in free space.

Solution:

1. Divergence Postulate (No Monopoles): ∇ ⋅ B = 0 ⟺ ∮ B ⋅ ds = 0
2. Curl Postulate (Ampere’s Law): ∇ × H = J ⟺ ∮ H ⋅ dl = I_enclosed

Question 2: Lorentz Force Equation

Write the Lorentz force equation for a charge q moving with velocity u in coexisting electric and magnetic fields.

Solution:

F = q(E + u × B)

Question 3: Biot-Savart Law vs. Ampere’s Law

When is it mathematically preferred to use Ampere’s Circuital Law rather than the Biot-Savart Law to calculate B?

Solution:

Ampere’s Law (∮ H ⋅ dl = I_enc) is preferred when the current distribution exhibits high spatial coordinate symmetry (infinite lines, cylinders, solenoids, toroids), allowing H to be factored out of the integral without explicit vector integrations.

Question 4: Field at Center of Circular Loop

Write the expression for magnetic flux density B at the center of a circular current loop of radius b carrying current I.

Solution:

B = a_z (μ₀ I / 2b)

Question 5: Vector Magnetic Potential Definition

Define Vector Magnetic Potential A and show why ∇ ⋅ B = 0 is automatically satisfied.

Solution:

Vector Magnetic Potential A is defined such that B = ∇ × A. Since the divergence of any vector curl is identically zero (∇ ⋅ (∇ × A) ≡ 0), taking the divergence yields ∇ ⋅ B = 0 automatically.

Question 6: Vector Poisson’s Equation

Write Vector Poisson’s equation for vector magnetic potential A in the Coulomb gauge (∇ ⋅ A = 0).

Solution:

∇² A = -μ₀ J

Question 7: Bound Current Densities

Define surface bound current density J_ms and volume bound current density J_m in terms of the magnetization vector M.

Solution:

• Surface Bound Current Density: J_ms = M × a_n
• Volume Bound Current Density: J_m = ∇ × M

Question 8: Normal Component Boundary Condition

State the boundary condition for the normal component of magnetic flux density B across an interface separating medium 1 (μ₁) and medium 2 (μ₂).

Solution:

B_1n = B_2n ⟹ μ₁ H_1n = μ₂ H_2n

The normal component of B is always continuous across any boundary.

Question 9: Tangential Component Boundary Condition

State the boundary condition for the tangential component of magnetic field intensity H across an interface carrying surface current density J_s.

Solution:

a_n2 × (H₁ - H₂) = J_s ⟹ H_1t - H_2t = J_s

If no free surface current exists (J_s = 0), H_1t = H_2t (tangential H is continuous).

Question 10: Hall Effect Voltage Formula

Write the formula for the Hall voltage V_H generated across a conductor strip of width w, carrier density n, carrying current I in magnetic field B.

Solution:

V_H = BI / nqw