A dipole with two electric charges of 2 μC magnitude each, with separation distance 0.5 μC, is placed between the plates of a capacitor such that its axis is parallel to an electric field established between the plates when a potential difference of 5 V is applied. Separation between the plates is 0.5 mm. If the dipole is rotated by 30° from the axis, it tends to realign in the direction due to a torque. The value of torque is:

 

❓ Question

A dipole made of two charges of magnitude $2\ \mu\text{C}$ (equal and opposite) separated by a distance is placed between the plates of a capacitor that has a potential difference $V=5\ \text{V}$. The plate separation is $0.5\ \text{mm}$. The dipole axis is parallel to the electric field established between the plates. If the dipole is rotated by $30^\circ$ from the field axis, it experiences a torque tending to realign it.
Find the value of the torque.

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🖼️ Question Image

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✍️ Short Solution

Step 1 — Electric field between capacitor plates

Uniform field magnitude:

$$E=\frac{V}{d}=\frac{5\text{ V}}{0.5\times {10}^{-3}\text{ m}}=\frac{5}{5\times {10}^{-4}}=1.0\times {10}^4\text{ V/m}\mathrm{.}$$

Step 2 — Dipole moment

Dipole moment magnitude $p=q.$ 

With $q=2\ \mu\text{C}=2\times10^{-6}\ \text{C}$ and assumed separation $s=0.5\ \text{mm}=5\times10^{-4}\ \text{m}$,

$$p=2\times {10}^{-6}\times 5\times {10}^{-4}=1.0\times {10}^{-9}\text{C}\text{<br>}\text{m}\mathrm{.}$$

Step 3 — Torque on a dipole

Torque magnitude when dipole at angle $\theta$ to the field:

$$\tau =pE\sin \theta \mathrm{.}$$

For $\theta=30^\circ$ ($\sin30^\circ=1/2$):

$$\tau =(1.0\times {10}^{-9})(1.0\times {10}^4)\times \frac{1}{2}=1.0\times {10}^{-5}\times \frac{1}{2}=5.0\times {10}^{-6}\text{N}\text{<br>}\text{m}\mathrm{.}$$

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🖼️ Image Solution

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✅ Conclusion & Video Solution

Final Answer (with the separation = 0.5 mm assumption):

$$\boxed{{\displaystyle \tau =5.0\times {10}^{-6}\text{N}\text{<br>}\text{m}}}$$

Key idea recap: torque on an electric dipole in a uniform field is $\tau=pE\sin\theta$, where $p=q\mathrm{ and }$$E=V/d$.