Derive an expression for the potential energy of an electric dipole placed in a uniform electric field.

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> [!TLDR]
> The potential energy of an electric dipole in a uniform field is found from the work needed to rotate it against the restoring torque, giving U(θ) = −pE cosθ = −**p**·**E**.
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> An electric dipole of moment $\vec p$ (magnitude $p = q	imes 2a$) placed in a uniform external electric field $\vec E$ experiences equal and opposite forces on its two charges (net force zero) but a net **torque**
> $$\vec	au = \vec p	imes\vec E,\qquad 	au = pE\sin	heta$$
> where θ is the angle between $\vec p$ and $\vec E$. This torque tends to rotate the dipole to align it with the field.
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> To rotate the dipole quasi-statically from angle $	heta_1$ to $	heta_2$ against this torque, external work must be done:
> $$W = \int_{	heta_1}^{	heta_2} 	au\,d	heta = \int_{	heta_1}^{	heta_2} pE\sin	heta\,d	heta = pE(\cos	heta_1 - \cos	heta_2)$$ …

Q.Derive an expression for the potential energy of an electric dipole placed in a uniform electric field.

Concept understanding — Electric Dipole in a Field

Electric Dipole in a Uniform Field — First Look

What happens? Two forces, one twist

The torque formula

What does this mean physically?

Q.Derive an expression for the potential energy of an electric dipole placed in a uniform electric field.

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