Question

An electric dipole of dipole moment \( p \) is kept in a uniform electric field \( E \) such that it is aligned parallel to the field. The energy required to rotate it by 45° is:

• A. \( pE \)
• B. \( pE \left(\frac{\sqrt{2} + 1}{\sqrt{2}}\right) \)
• C. \( pE \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right) \)
• D. \( \frac{pE}{\sqrt{2}} \)
• E. \( \sqrt{2} pE \)

Hint:

For a dipole in a uniform electric field, the energy change during rotation depends on the cosine of the angle between the dipole moment and the electric field.

Answer 1

The Correct Option is C

Step 1: The potential energy of a dipole in an electric field is given by: \[ U = -pE \cos\theta \] where: 
- \( p \) is the dipole moment, - \( E \) is the electric field, 
- \( \theta \) is the angle between the dipole moment and the electric field. 
Step 2: The change in potential energy when the dipole is rotated by 45° is: \[ \Delta U = U(\theta = 45^\circ) - U(\theta = 0^\circ) \] Substituting \( \theta = 45^\circ \) and \( \theta = 0^\circ \): \[ \Delta U = -pE \cos 45^\circ + pE \cos 0^\circ = -pE \left(\frac{1}{\sqrt{2}}\right) + pE = pE \left(1 - \frac{1}{\sqrt{2}}\right) \] 
Step 3: Simplifying: \[ \Delta U = pE \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right) \] 
Thus, the energy required to rotate the dipole by 45° is \( pE \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right) \).