Depict the orientation of an electric dipole in (a) stable and (b) unstable equilibrium in an external uniform electric field. Write the potential energy of the dipole in each case.

(a) Write the expression for the Lorentz force on a particle of charge $$q$$ moving with a velocity $$\vec{v}$$ in a magnetic field $$\vec{B}$$. When is the magnitude of this force maximum? Show that no work is done by this force on the particle during its motion from a point $$\vec{r_1} \text { to point } \vec{r}_2 \text {. }$$

OR

(b) A long straight wire $$A B$$ carries a current I. A particle (mass $$m$$ and charge $$q$$ ) moves with a velocity $$\vec{v}$$, parallel to the wire, at a distance $$d$$ from it as shown in the figure. Obtain the expression for the force experienced by the particle and mention its directions.

The potential difference applied across a given conductor is doubled. How will this affect (i) the mobility of electrons and (ii) the current density in the conductor? Justify your answers.

Two coils $$C_1$$ and $$C_2$$ are placed close to each other. The magnetic flux $$\phi_2$$ linked with the coil $$\mathrm{C}_2$$ varies with the current $$I_1$$ flowing in coil $$C_1$$, as shown in the figure. Find

(i) the mutual inductance of the arrangement, and

(ii) the rate of change of current $$\left(\frac{d \mathrm{I}_1}{d t}\right)$$ that will induce an emf of 100 V in coil C$$_2$$.