(i) Find the value of the phase difference between the current and the voltage in the series LCR circuit shown below. Which one leads in phase : current or voltage ? (ii) Without making any other change, find the value of the additional capacitor C₁, to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.

(i) When an ac source is connected to an ideal inductor shows that the average power supplied by the source over a complete cycle is zero. (ii) A lamp is connected in series with an inductor and an ac source. What happens to the brightness of the lamp when the key is plugged in and an iron rod is inserted inside the inductor ? Explain.

A sinusoidal voltage of peak value 10 V is applied to a series LCR circuit in which resistance,capacitance and inductance have values of 10 Ω,1 µF and 1 H respectively. Find (i) the peak voltage across the inductor at resonance (ii) quality factor of the circuit.

(i) A point charge (+Q) is kept in the vicinity of an uncharged conducting plate. Sketch electric field lines between the charge and the plate. (ii) Two infinitely large plane thin parallel sheets having surface charge densities σ₁ and σ₂ (σ₁ > σ₂) are shown in the figure. Write the magnitudes and directions of the fields in the regions marked II and III.

(i) An electric dipole is kept first to the left and then to the right of a negatively charged infinite plane sheet having a uniform surface charge density. The arrows p₁ and p₂ show the directions of its electric dipole moments in the two cases.

A source of ac voltage V = V₀ sin ωt, is connected across a pure inductor of inductance L. Derive the expressions for the instantaneous current in the circuit. Show that average power dissipated in the circuit is zero.

An electric dipole is placed in a uniform electric field. (i) Show that no translatory force acts on it. (ii) Derive an expression for the torque acting on it. (iii) Find work done in rotating the dipole through 180°.

(i) Three point charges q, – 4q and 2q are placed at the vertices of an equilateral triangle ABC of side ‘l’ as shown in the figure. Obtain the expression for the magnitude of the resultant electric force acting on the charge q.

A particle, having a charge +5 µC, is initially at rest at the point x = 30 cm on the x-axis. The particle begins to move due to the presence of a charge Q that is kept fixed at the origin. Find the kinetic energy of the particle at the instant it has moved 15 cm from its initial position if (a) Q = +15 µC and (b) Q = – 15 µC

A charge + Q, is uniformly distributed within a sphere of radius R. Find the electric field, due to this charge distribution, at a distant point r from the centre of the sphere where : (i) 0 < r < R (ii) r > R

A charge is distributed uniformly over a ring of radius ‘a’. Obtain an expression for the electric intensity E at a point on the axis of the ring. Hence, show that for points at large distances from the ring, it behaves like a point charge.

Derive an expression for electric field of a dipole at a point on the equatorial plane of the dipole. How does the field vary at large distances?

Four point charges Q, q, Q and q are placed at the corners of a square of side ‘a’ as shown in the figure.

Obtain the expression for the potential due to an electric dipole of dipole moment p at a point ‘d’ on the axial line.

Two point charges + q and –2q are placed at the vertices ‘B’ and ‘C’ of an equilateral triangle ABC of side ‘a‘ as given in the figure. Obtain the expression for (i) the magnitude and (ii) the direction of the resultant electric field at the vertex A due to these two charges.