Torque on Electric Dipole in Uniform Electric Field

 

❓ 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 Explanation

This problem is based on:

👉 Electric dipole
👉 Electric field in capacitor
👉 Torque on dipole.

Main idea:

Torque on dipole:

$$\boxed{ \tau=pE\sin\theta }$$

where:

$$p=ql$$

and electric field between capacitor plates:

$$\boxed{ E=\frac{V}{d} }$$

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🔷 Step 1 — Find Electric Field 💯

Given:

$$V=5\text{ V}$$

Plate separation:

$$d=0.5\text{ mm} =0.5\times10^{-3}\text{ m}$$

Electric field:

$$E=\frac{V}{d}$$
$$= \frac{5}{0.5\times10^{-3}}$$
$$=10^4\ \text{N/C}$$

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🔷 Step 2 — Find Dipole Moment

Charge:

$$q=2\mu C =2\times10^{-6}\text{ C}$$

Separation:

$$l=0.5\mu m =0.5\times10^{-6}\text{ m}$$

Dipole moment:

$$p=ql$$
$$= (2\times10^{-6}) (0.5\times10^{-6})$$
$$=10^{-12}\ \text{C m}$$

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🔷 Step 3 — Calculate Torque

Formula:

$$\tau=pE\sin\theta$$

Given:

$$\theta=30^\circ$$
$$\sin30^\circ=\frac12$$

Substitute:

$$\tau = (10^{-12}) (10^4) \left(\frac12\right)$$
$$= 5\times10^{-9}\text{ Nm}$$

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🔷 Step 4 — JEE Trap Alert 🚨

❌ $\mu m$ and $mm$ conversion mistake

❌ Dipole length ko cm me convert kar dena

❌ $\sin30^\circ$ factor miss kar dena

Remember:

$$\boxed{ \tau=pE\sin\theta }$$

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✅ Final Answer

$$\boxed{ 5\times10^{-9}\ \text{Nm} }$$

(Option 2)

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