An electric dipole consists of two point charges +q and -q having masses m and 3m respectively, separated by a distance d . It is placed in a uniform electric field E . When slightly displaced from its equilibrium position and released, it executes small oscillations with a time period T . If the magnitude of the electric field is given by E = x ^2 m d q T^2 , the value of x is:

Step-by-step solution

The dipole is free to rotate, so it will oscillate about its center of mass. The moment of inertia of the two-particle system about its center of mass is calculated using the reduced mass : = m 3m m + 3m = 3m 4 The moment of inertia is: I = d^2 = 3 4 md^2 The restoring torque for a small angular displacement is: = -pE -pE where the dipole moment is p = qd . The time period of small oscillations is given by: T = 2 I pE Substituting the values of I and p : T = 2 3 4 md^2 qdE = 2 3md 4qE Squaring both sides: T^2 = 4 ^