(a) Deduce the expression for the torque acting on a dipole of dipole moment in the presence of a uniform electric field .
(b) Consider two hollow concentric spheres, S1 and S2, enclosing charges 2Q and 4Q respectively as shown in the figure.
(i) Find out the ratio of the electric flux through them.
(ii) How will the electric flux through the sphere S1 change if a medium of dielectric constant 'εr' is introduced in the space inside S1 in place of air ? Deduce the necessary expression.
Answers
T→ = p→ * E→
Ratio of electric flux = 1:3
Electric flux decreases
Explanation:
(a) Dipole is a uniform electric field.
Let an electric dipole with charges -q and +q and of length 2a be placed in a uniform electric field at an angle θ.
Forces acting on the charges will be +qE and -qE. The net force will be equal and opposite. So force = 0.
Two forces are equivalent to torque having magnitude,
T = (qE)AC = qE.2a sinθ = pEsinθ
Torque is given by T→ = p→ * E→
(b) (i) Charge enclosed by sphere S1 = 2Q
Charge enclosed by sphere S2 = 2Q + 4Q = 6Q
As per Gauss Law, electric flux enclosed by both sphere is:
∅r = 2Q/ϵo and ∅2 = 6Q/ϵo
Ratio of electric flux = ∅1/ ∅2 = (2Q/ϵo) / (6Q/ϵo) = 1/3
(ii) If a medium of dielectric constant ϵr is introduced inside S1 instead of air, electric flux will become:
∅r = 2Q/ϵr where ϵr > ϵo
Therefore, ∅r < ∅1 meaning electric flux decreases.