Electric Charges and Fields

An infinite sheet carrying a uniform surface charge density σ lies on the xy-plane. The work done to carry a charge q from the point $\mathrm{A} = \mathrm{a}(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}})$ to the point $\mathrm{B}\hspace{0.17em}= \mathrm{a}(\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}})$ (where a is a constant with the dimension of length and ε₀ is the permittivity of free space) is

• $\frac{3\mathrm{\sigma aq}}{2{\mathrm{\epsilon }}_0}$

• $\frac{2\mathrm{\sigma aq}}{{\mathrm{\epsilon }}_0}$

• $\frac{5\mathrm{\sigma aq}}{2{\mathrm{\epsilon }}_0}$

• $\frac{3\mathrm{\sigma aq}}{{\mathrm{\epsilon }}_0}$

Consider two concentric spherical metal shells of radii r₁ and r₂ (r₂ > r₁). If the outer shell has a charge q and the inner one is grounded, the charge on the inner shell is

• $\frac{- {\mathrm{r}}_2}{{\mathrm{r}}_1} \mathrm{q}$

• zero

• $\frac{- {\mathrm{r}}_1}{{\mathrm{r}}_2} \mathrm{q}$

• - q

The line AA' is on charged infinite conducting plane which is perpendicular to the plane of the of the paper. The plane has a surface density of charge σ and B is ball of mass m with a like charge of magnitude q. B is connected by string from a point on the line AA'. The tangent of angle (θ) formed between the line AA' and the string is

• $\frac{\mathrm{q\sigma }}{2{\mathrm{\epsilon }}_0\mathrm{mg}}$

• $\frac{\mathrm{q\sigma }}{4{\mathrm{\pi \epsilon }}_0\mathrm{mg}}$

• $\frac{\mathrm{q\sigma }}{2{\mathrm{\pi \epsilon }}_0\mathrm{mg}}$

• $\frac{\mathrm{q\sigma }}{{\mathrm{\epsilon }}_0\mathrm{mg}}$

A hollow sphere of external radius R and thickness (t << R) is made of a metal of density ρ. The sphere will float in water, if

• $\mathrm{t} \le \frac{\mathrm{R}}{\mathrm{\rho }}$

• $\mathrm{t} \le \frac{\mathrm{R}}{3\mathrm{\rho }}$

• $\mathrm{t} \le \frac{\mathrm{R}}{2\mathrm{\rho }}$

• $\mathrm{t} \ge \frac{\mathrm{R}}{3\mathrm{\rho }}$

A charge q is placed at one corner of a cube. The electric flux through any of the three faces adjacent to the charge is zero. The flux through any one of the other three faces is

• q/3ε₀

• q/6ε₀

• q/12ε₀

• q/24ε₀

Two charges + q and -q are placed at a distance a in a uniform electric field. The dipole moment of the combination is $2\mathrm{qa} \left(\cos \mathrm{\theta } \hat{\mathrm{i}} + \sin \mathrm{\theta } \hat{\mathrm{j}}\right)$, where, θ is the angle between the direction of the field and the line joining the two charges. Which of the following statement(s) is/are correct ?

• The torque exerted by the field on the dipole vanishes

• The net force on the dipole vanishes

• The torque is independent of the choice of coordinates

• The net force is independent of a