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Step-by-Step Solution
Step 1: Understand the Problem
We have an electric dipole of moment $ \overrightarrow{p} $ placed in a uniform electric field $ \overrightarrow{E} $. Initially, the dipole is aligned along the electric field. We need to find the work done to rotate this dipole by $90^\circ$ from its initial alignment.
Step 2: Recall the Expression for Work Done
The work done $W$ in rotating the dipole from an angle $0$ to an angle $\theta$ with respect to the field is given by the integral:
$$
W \;=\; \int_{0}^{\theta} p E \,\sin\alpha \; d\alpha
$$
where $p$ is the magnitude of the electric dipole moment, and $E$ is the magnitude of the uniform electric field.
Step 3: Evaluate the Integral
To find $W$, perform the integration:
$$
W = pE \int_{0}^{\theta} \sin\alpha \; d\alpha.
$$
$$
= pE \left[-\cos\alpha\,\right]_{0}^{\theta}.
$$
$$
= pE \Bigl[\,-(\cos\theta - \cos 0)\Bigr].
$$
$$
= pE \bigl[\,\cos 0 - \cos\theta\bigr].
$$
$$
= pE \left( 1 - \cos\theta \right).
$$
Step 4: Substitute $\theta = 90^\circ$
At $90^\circ$, $\cos 90^\circ = 0$. Therefore,
$$
W = pE\left(1 - \cos 90^\circ\right) = pE\bigl(1 - 0\bigr) = pE.
$$
Step 5: State the Final Answer
Hence, the work done in rotating the dipole by $90^\circ$ in a uniform electric field is
$$
\boxed{pE}.
$$
