Aspherical conductor of radius a is placed in a uniform electric field $$\overrightarrow {\bf{E}} = {E_0}\,{\bf{\hat k}}.$$   The potential at a point P(r, θ) for r > a, is given by $$\phi \left( {r,\,\theta } \right) = {\text{constant}} - {E_0}r\cos \theta + \frac{{{E_0}{a^3}}}{{{r^2}}}\cos \theta $$
where, r is the distance of P from the centre O of the sphere and θ is the angle, OP makes with the Z-axis.

The charge density on the sphere at θ = 30° is

Which one of the following current densities, $$\overrightarrow {\bf{J}} $$ can generate the magnetic vector potential $$\overrightarrow {\bf{A}} = \left( {{y^2}{\bf{\hat i}} + {x^2}{\bf{\hat j}}} \right)?$$

A. $$\frac{2}{{{\mu _0}}}\left( {x{\bf{\hat i}} + y{\bf{\hat j}}} \right)$$

B. $$ - \frac{2}{{{\mu _0}}}\left( {{\bf{\hat i}} + {\bf{\hat J}}} \right)$$

C. $$ - \frac{2}{{{\mu _0}}}\left( {{\bf{\hat i}} - {\bf{\hat j}}} \right)$$

D. $$\frac{2}{{{\mu _0}}}\left( {x{\bf{\hat i}} - y{\bf{\hat j}}} \right)$$

A metal has free-electron density n = 10²⁹ m^-3. Which of the following wavelengths will excite plasma oscillations?

A. 0.033 μm

B. 0.330 μm

C. 3.300 μm

D. 33.000 μm

The electric field of a plane electromagnetic wave is $$\overrightarrow {\bf{E}} = \overrightarrow {{{\bf{E}}_0}} \exp \left[ {i\left( {{\bf{\hat x}}k\cos \alpha + {\bf{\hat y}}k\sin \alpha - \omega t} \right)} \right].$$        If $${\bf{\hat x}},\,{\bf{\hat y}}$$  and $${{\bf{\hat z}}}$$ are cartesian unit vectors, the wave vector $$\overrightarrow {\bf{k}} $$ of the electromagnetic wave is

A. $${\bf{\hat z}}k$$

B. $${\bf{\hat x}}k\sin \alpha + {\bf{\hat y}}k\cos \alpha $$

C. $${\bf{\hat x}}k\cos \alpha + {\bf{\hat y}}k\cos \alpha $$

D. $$ - {\bf{\hat z}}k$$

The dispersion relation for a, low density plasma is ω² = ω₀² + c² k², where ω₀ is the plasma frequency and c is the speed of light in free space. The relationship between the group velocity (v_g) and phase velocity (v_p) is

A. v_p = v_g

B. v_p = $${\text{v}}_{\text{g}}^{\frac{1}{2}}$$

C. v_p v_g = c²

D. v_g = $${\text{v}}_{\text{p}}^{\frac{1}{2}}$$