To rotate a dipole, work needs to be done against the electric field. The work done to rotate a dipole from an initial angle of 30° to a final angle of 90° can be calculated using the formula:

Work = -ΔPE = -qΔV

Where q is the charge on the dipole and ΔV is the change in potential energy.

Explanation:
1. Electric Field of a Dipole:
A dipole consists of two equal and opposite charges separated by a distance. The electric field produced by a dipole at a point on its axial line is given by:

E = (kp / r^3) * (2cosθ)

Where E is the electric field, kp is the electrostatic constant, r is the distance from the center of the dipole, and θ is the angle between the dipole axis and the line joining the point and the center of the dipole.

2. Potential Energy of a Dipole:
The potential energy of a dipole in an electric field is given by:

PE = -pE

Where PE is the potential energy, p is the dipole moment, and E is the electric field.

3. Initial and Final Angles:
In this case, the initial angle is 30° and the final angle is 90°. The change in potential energy can be calculated as:

ΔPE = PE(final) - PE(initial)

4. Calculating Work:
Using the formula for potential energy, the change in potential energy can be expressed as:

ΔPE = -pE(final) - (-pE(initial))
ΔPE = p(E(initial) - E(final))

Substituting the expressions for electric field, the change in potential energy becomes:

ΔPE = p * [ (kp / r^3) * (2cosθ(initial)) - (kp / r^3) * (2cosθ(final)) ]

5. Work Done:
Finally, the work done to rotate the dipole can be calculated using the formula mentioned earlier:

Work = -ΔPE = -qΔV
Work = -p * [ (kp / r^3) * (2cosθ(initial)) - (kp / r^3) * (2cosθ(final)) ]

This is the detailed explanation of the work done to rotate a dipole from an initial angle of 30° to a final angle of 90°.