Question

An electric dipole consists of two point charges \(+1\,\mu\text{C}\) and \(-1\,\mu\text{C}\), held 10 cm apart. It is subjected to a uniform electric field of \(100\,\text{N/C}\). Calculate the amount of work done in turning the dipole from its position of stable equilibrium to the position of unstable equilibrium, in the field.

Hint:

For rotation of a dipole from stable equilibrium \((0^\circ)\) to unstable equilibrium \((180^\circ)\), \[ W=2pE \] This shortcut directly gives the required work done.

Answer 1

Concept: An electric dipole placed in a uniform electric field possesses potential energy due to its orientation in the field. The potential energy of an electric dipole in a uniform electric field is given by \[ U=-pE\cos\theta \] where \[ p=q(2l) \] is the dipole moment, \[ E \] is the magnitude of the electric field and \[ \theta \] is the angle between the dipole moment vector and the electric field. The work done by an external agent in rotating the dipole slowly from one orientation to another is equal to the increase in its potential energy. \[ W=\Delta U \] The stable equilibrium position corresponds to minimum potential energy, while the unstable equilibrium position corresponds to maximum potential energy.

Step 1: Calculate the dipole moment. Magnitude of each charge is \[ q=1\,\mu C =1\times10^{-6}\,C \] Distance between the charges is \[ d=10\,cm =0.10\,m \] Therefore, \[ p=qd \] \[ p=(1\times10^{-6})(0.10) \] \[ p=1\times10^{-7}\,C\,m \]

Step 2: Determine the potential energy in stable equilibrium. Stable equilibrium occurs when the dipole aligns parallel to the electric field. Hence, \[ \theta=0^\circ \] Using \[ U=-pE\cos\theta \] we get \[ U_s=-pE\cos0^\circ \] \[ U_s=-pE \] Substituting values, \[ U_s=-(1\times10^{-7})(100) \] \[ U_s=-10^{-5}\,J \]

Step 3: Determine the potential energy in unstable equilibrium. Unstable equilibrium occurs when the dipole is antiparallel to the electric field. Thus, \[ \theta=180^\circ \] Hence, \[ U_u=-pE\cos180^\circ \] \[ U_u=-pE(-1) \] \[ U_u=+pE \] \[ U_u=(1\times10^{-7})(100) \] \[ U_u=10^{-5}\,J \]

Step 4: Calculate the work done. Work done by the external agent equals the increase in potential energy. \[ W=U_u-U_s \] \[ W=(10^{-5})-(-10^{-5}) \] \[ W=2\times10^{-5}\,J \]

Step 5: Write the final answer. Therefore, the work required to rotate the dipole from stable equilibrium to unstable equilibrium is \[ \boxed{2\times10^{-5}\,J} \]