Derive the formula for the torque acting on a dipole placed in a … - Vidyadip

Question

Derive the formula for the torque acting on a dipole placed in a uniform external electric field.

✓

Answer

→ As shown in figure, the electric dipole is placed in uniform electric field at $\theta$ angle.
→The force exerted on $+q$ electric charge in electric field $\overrightarrow{ E }_{\text {is }}$,
$
\overrightarrow{F}+=q \overrightarrow{E}
$
→ The force exerted on - $q$ electric charge
$
\overrightarrow{F}_{-}=-q \overrightarrow{E}
$
→ The net force on the dipole is zero, since $E$ is uniform.
→ However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole.
→ When the net force is zero, the torque (couple) is independent of the origin. Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).
→ Magnitude of torque $=q E \times 2 a \sin \theta$
$
\begin{array}{l}
=(2 q a) E \sin \theta=p E \sin \theta \\
\vec{\tau}=\vec{p} \cdot \overrightarrow{E}
\end{array}
$
→ The magnitude of $\vec{p} \times \overrightarrow{ E }$ is also $p E \sin \theta$ and its direction is normal to the paper, coming out of it.
→ Special cases :
(i) If electric dipole moment and electric field both are in one direction.
$
\begin{array}{l}
\therefore \theta=0 \\
\therefore \tau=0
\end{array}
$
This condition is called stable equilibrium.
(ii) Both electric dipole moment and electric field are perpendicular.
$
\therefore \theta=\frac{\pi}{2}
$
$\therefore \tau=p E \sin \frac{\pi}{2}$
$\therefore \tau=p E \rightarrow$ Which is maximum
(iii) Electric dipole moment and electric field are arranged anti parallel direction.
$
\begin{array}{l}
\therefore \tau=pE \sin \pi \\
\therefore \tau=0 \rightarrow \text { Unstable equilibrium }
\end{array}
$

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