(i) An electric dipole is placed at an angle of 30° with an electric field of intensity

2 × 10⁵NC^–1. It experiences a torque of 4Nm. Calculate the charge on the dipole if the dipole length is 2 cm

(ii) If a charge q is placed at the centre of the line joining two equal charges Q such that the system is in equilibrium then what is the value of q?

OR

A circular coil of mean radius rand having N turns is kept in a horizontal plane. A magnetic field B exists in the vertical direction as shown in figure. Find the emf induced in the loop

(a) if it is held stationary and the magnetic field is uniform,

(b) if it is held stationary and the magnetic field is non-uniform,

(c) if it is rotated with an angular velocity co about an axis passing through its centre and perpendicular to its plane, and the magnetic field is uniform.

(d) if it is rotated with an angular velocity co about its diameter. Assume that the normal to the plane of the coil makes an angle  = 0 with the magnetic field at time t = 0.

(e) if the coil is a square of side L and is rotated about its diameter.

Torque,  $$\begin{gathered} \overrightarrow{\mathrm{\tau }}=\overrightarrow{\mathrm{p}}\times \overrightarrow{\mathrm{E}}=\mathrm{pEsin} \mathrm{\theta } \\ 4=\mathrm{p}\times 2\times {10}^5\times \sin {30}^{\circ } \end{gathered}$$

or $\mathrm{p}=\frac{4}{2\times {10}^5\times \sin {30}^{\circ }}=4\times {10}^{-5}\mathrm{Cm}$

Dipole moment, $\mathrm{p}=\mathrm{q}\times 1$

$\mathrm{q}=\frac{\mathrm{p}}{l}=\frac{4\times {10}^{-5}}{0.02}=2\times {10}^{-3}\mathrm{C}=2\mathrm{mC}$

(ii) Let q charge is situated at the mid position of the line AB. The distance between AB is x. A and B be the positions of charges Q and Q respectively

Let $\mathrm{AC}=\frac{\mathrm{x}}{2},\mathrm{BC}=\frac{\mathrm{x}}{2}$

The force on A due to charge q at C.

${\overrightarrow{\mathrm{F}}}_{\mathrm{CA}}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\cdot \frac{\mathrm{Q}\cdot \mathrm{q}}{(\mathrm{x}/2{)}^2}$ along $\overrightarrow{\mathrm{AC}}$

The force on A due to charge Q at B

${\overrightarrow{\mathrm{F}}}_{\mathrm{AB}}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\cdot \frac{{\mathrm{Q}}^2}{{\mathrm{x}}^2} \mathrm{along} \overrightarrow{\mathrm{BA}}$

The system is in equilibrium, then two oppositely directed force must be equal, i.e., total force on A is equal to zero.

$\begin{array}{rr} & {\overrightarrow{\mathrm{F}}}_{\mathrm{CA}}+{\overrightarrow{\mathrm{F}}}_{\mathrm{AB}}=0\Rightarrow {\overrightarrow{\mathrm{F}}}_{\mathrm{CA}}=-{\overrightarrow{\mathrm{F}}}_{\mathrm{AB}}\\ & \frac{1}{4\pi {\epsilon }_0}\cdot \frac{4\mathrm{Q}\cdot \mathrm{q}}{{\mathrm{x}}^2}=\frac{-1}{4\pi {\epsilon }_0}\cdot \frac{{\mathrm{Q}}^2}{{\mathrm{x}}^2}\\ & \Rightarrow \mathrm{q}=-\frac{\mathrm{Q}}{4}\end{array}$

OR

The emf is induced .if the magnetic flux through the coil changes with time.

(a)   In this case there is no change in magnetic flux with time, hence no emf induced.

(b)   In this case also the magnetic flux through the coil does not change with time, hence no emf induced.

(b)   In this case, the number of field lines through the coil does not change with time, hence the magnetic flux does not change with time. So no emf is induced in the coil

(d) If the coil is rotated about a diameter, as shown in figure, there is a change in magnetic flux with time. Hence emf will be induced in the coil. Area of the coil is A = pr². If the normal to the plane of the coil makes an angle $\theta$ = 0 with the magnetic field, at . times t = 0 then at time t, $\theta$ = wt. The magnetic flux at this time is

$\mathrm{\Phi }=\mathrm{NBAcos}\theta =\mathrm{NBAcos}\mathrm{wt}$

:. Induced emf is

$\begin{array}{rr}\mathrm{e}& \mathrm{ }=-\frac{\mathrm{d\varphi }}{\mathrm{dt}}=-\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{NBAcos}\mathrm{\omega t})\\ & \mathrm{ }=\mathrm{\omega NBAsin}\mathrm{\omega t}\\ & \mathrm{ }=\mathrm{\omega NB}\times {\mathrm{\pi r}}^2\sin \mathrm{\omega t}\\ & \mathrm{ }={\mathrm{\pi Nr}}^2\mathrm{B\omega sin}\mathrm{\omega t}\end{array}$

(e) If a square coil of side L and N turns is rotated about a diameter as shown in figure, there is a change in magnetic flux with time. Hence emf will be induced in the coil. Area of the coil is A = L²

If the normal to the plane of the coil makes an angle $\theta$ = 0 with-the magnetic field at time t = 0, then at time t, $\theta$ = wt.

The magnetic flux at this time is

Φ = NBA cos $\theta$ = NB L² cos wt

:. Induced emf is

$e=-\frac{d\varphi }{dt}=NB{L}^2\omega \sin \left(\omega t\right)$

Thus, as in case (d) above, an alternating emf is induced in the coil.