4. The potential at the surface of a sphere (radius R) is given by V₀=k cos3θ where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density σ(θ) on the sphere. (Assume there's no charge inside or outside the sphere.)
(A)(ε₀k/6R)cosθ[140cos²θ - 93]
(B)(ε₀k/5R)cosθ[140cos²θ]
(C)(ε₀k/5R)cosθ[140cosθ - 93]
(D)(ε₀k/5R)cosθ[140cos²θ - 93]

1. Find the angle between the body diagonals of a cube. (A)θ=cos⁻¹(1/3) (B)θ=cos⁻¹(1/√3) (C)θ=cos⁻¹(1/2) (D)θ=cos⁻¹(1/√2)

2. Find the divergence of the function v = s(2+sin²φ)ŝ + s sinφ cosφ φ̂ + 3zẑ (A)5 (B)10 (C)8 (D)3

3. One of these is a possible electrostatic field. Which one? (A)E = k[xyx̂ + 2yzŷ + 3xzẑ] (B)E = k[y²x̂ + (2xy + z²)ŷ + 2yzẑ] (C)E = k[yx̂ + x²ẑ] (D)E = k[yzx̂ + y²ŷ + z²ẑ]

5. A certain coaxial cable consists of a copper wire, radius a, surrounded by a concentric copper tube of inner radius c. The space between is partially filled (from b out to c) with material of dielectric constant εᵣ, as shown. Find the capacitance per unit length of this cable. (A)2πε₀/[ln(a/b) + (1/εᵣ)ln(c/b)] (B)2πε₀/[ln(b/a) + (1/ε)ln(c/b)] (C)2πε₀/[ln(b/a) + (1/εᵣ)ln(c/b)] (D)2πε₀/[ln(b/a) + (1/εᵣ)ln(b/c)]

6. A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field outside the wire if the current is uniformly distributed over the outside surface of the wire. (A)0 (B)μ₀Is²/(2πa³) φ̂ (C)μ₀Is²/(2πa²) φ̂ (D)μ₀I/(2πs) φ̂

7. An infinitely long circular cylinder carries a uniform magnetization M parallel to its axis. Find the magnetic field (due to M) inside the cylinder. (A)2μ₀M (B)μ₀M (C)0 (D)μ₀M/4π

8. A square loop of wire, of side a, lies midway between two long wires, 3a apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clockwise current I in the square loop is gradually increasing: dI/dt = k (a constant). Find the emf induced in the big loop. Which way will the induced current flow? (A)μ₀ka ln2/π, counterclockwise (B)μ₀ka ln2/π, clockwise (C)μ₀ka ln2/(2π), counterclockwise (D)μ₀ka ln2/(2π), clockwise

9. A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width ω≪a, forms a parallel-plate capacitor, as shown in figure. Find the magnetic field in the gap, at a distance s ＜ a from the axis. (A)μ₀Is/(πa²) φ̂ (B)μ₀Is³/(2πa) φ̂ (C)μ₀Is/(2πa²) φ̂ (D)μ₀Ia/(2π) φ̂

10. Express the wave equation of the standing wave f(z,t) = A sin(kz)cos(kvt) as the sum of a wave traveling to the left and a wave traveling to the right. (A)(A/2){sin[k(z+vt)] + sin[k(z-vt)]} (B)(A/2){cos[k(z+vt)] + sin[k(z-vt)]} (C)(A/2){sinh[k(z+vt)] + sinh[k(z-vt)]} (D)(A/2){cosh[k(z+vt)] + cosh[k(z-vt)]}

20. 一平行板電容器帶電Q及-Q，兩板相距d，兩板間電場數值為E，則其單位面積儲存的電場能為？ (A) (B) (C)(D) (E)ε₀E²

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