An electric dipole with a dipole moment p=4\times 10^{-9},\text{C·m} is placed in a uniform electric field of magnitude E=3\times 10^{3},\text{N/C} such that the dipole is in stable equilibrium. Stable Equilibrium Condition:
- The dipole is in stable equilibrium when its dipole moment \mathbf{p} is aligned parallel to the electric field \mathbf{E}, i.e., the angle \theta =0^\circ between \mathbf{p} and \mathbf{E}
Work Done in Rotating Dipole from Stable to Unstable Equilibrium:
- The potential energy of a dipole in an electric field is given by:
U=-pE\cos \theta
- At stable equilibrium (\theta =0^\circ ):
U_{\text{stable}}=-pE
- At unstable equilibrium (\theta =180^\circ ):
U_{\text{unstable}}=-pE\cos 180^\circ =pE
- The work done W to rotate the dipole from stable to unstable equilibrium is the change in potential energy:
W=U_{\text{unstable}}-U_{\text{stable}}=pE-(-pE)=2pE
- Substituting values:
W=2\times (4\times 10^{-9})\times (3\times 10^{3})=2.4\times 10^{-5},\text{J}
Summary:
- The dipole is stable when aligned with the field (\theta =0^\circ ).
- The work done to rotate it to the unstable equilibrium (\theta =180^\circ ) is 2.4\times 10^{-5},\text{J}
Thus, the dipole in stable equilibrium aligns parallel to the electric field, and the work done to rotate it to the unstable position is 2.4\times 10^{-5},\text{J}.
