An electric dipole consists of two point charges +q and -q having masses m and 3m respectively, separated by a distance L . It is placed in a uniform electric field E . In Case 1, the dipole is completely free to rotate. In Case 2, the dipole is hinged at the positive charge ( +q ) so that it can only rotate about that point. If f₁ and f₂ are the frequencies of small oscillations of the dipole in Case 1 and Case 2 re

Step-by-step solution

For Case 1 (free dipole): The dipole rotates about its center of mass. The moment of inertia is calculated using the reduced mass : = m 3m m + 3m = 3m 4 The moment of inertia about the center of mass is: I₁ = L^2 = 3 4 mL^2 The restoring torque is = -pE = -qLE . The frequency of oscillation is: f₁ = 1 2 qLE I₁ = 1 2 qLE 3 4 mL^2 = 1 2 4qE 3mL For Case 2 (hinged at +q ): The dipole rotates about the fixed hinge at +q . The moment of inertia about this axis is due only to the mass of -q : I₂ = (3m)L^2 = 3mL^2 The ele