Q.An electric dipole with dipole moment $4 \times 10^{-9}\,\text{C m}$ is aligned at $30^\circ$ with the direction of a uniform electric field of magnitude $5 \times 10^{4}\,\text{N C}^{-1}$. Calculate the magnitude of the torque acting on the dipole.

Concept: Torque on an Electric Dipole

When an electric dipole of moment $\vec{p}$ is placed in a uniform electric field $\vec{E}$, it experiences a torque that tries to align it with the field. The magnitude of this torque is given by

where $\theta$ is the angle between the dipole moment and the electric field direction.

Calculation:

Given:

• Dipole moment: $p = 4 \times 10^{-9}\,\text{C m}$
• Electric field: $E = 5 \times 10^{4}\,\text{N C}^{-1}$
• Angle: $\theta = 30^\circ$

Substituting into the torque formula: …

Why a dipole experiences torque

An electric dipole consists of two equal and opposite charges separated by a small distance. When placed in a uniform electric field, both charges experience forces of equal magnitude but in opposite directions. Because the charges are spatially separated, these forces don't simply cancel—they create a couple that tries to rotate the dipole.

The key insight is that the torque depends on how misaligned the dipole is with the field. When the dipole moment vector $\vec{p}$ points along the field $\vec{E}$, the forces on both charges lie along the dipole axis and produce no rotation. When perpendicular, the lever arm is maximum and torque peaks. At any intermediate angle $\theta$, only the component of force perpendicular to the dipole axis contributes to rotation.

where $p$ is the dipole moment magnitude, $E$ is the field strength, and $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

Step-by-step calculation

1. Identify the given quantities

   • Dipole moment: $p = 4 \times 10^{-9}\,\text{C m}$
   • Electric field: $E = 5 \times 10^{4}\,\text{N C}^{-1}$
   • Angle between dipole and field: $\theta = 30°$

2. Recognize the torque formula

   The magnitude of torque on a dipole in a uniform field is the cross-product magnitude:

   $\tau = |\vec{p} \times \vec{E}| = pE\sin\theta$ …

Method: Torque from the Potential Energy Function

A dipole in a uniform field doesn't just feel a torque — it has an orientation-dependent potential energy. Torque is nothing but how fast that energy changes as you rotate the dipole, which gives an equivalent, more general way to arrive at the same result.

1. Write down the potential energy of the dipole. When a dipole moment $\vec{p}$ makes angle $\theta$ with a uniform field $\vec{E}$, its potential energy is

This is lowest (most stable) when $\vec{p}$ is aligned with $\vec{E}$ ($\theta = 0$) and highest when anti-aligned ($\theta = 180°$) — exactly what we'd expect physically.

2. Recall the rotational analogue of $F = -\dfrac{dU}{dx}$. For rotation, the torque about an axis is the negative derivative of potential energy with respect to the rotation angle:

3. Differentiate. $\tau = -\frac{d}{d\theta}\left(-pE\cos\theta\right) = pE\sin\theta$ …

Step 1 — The Correct Formula

The torque $\vec{\tau}$ on an electric dipole in a uniform electric field $\vec{E}$ is:

Magnitude:

Where:

• $p$ = dipole moment magnitude
• $E$ = electric field magnitude
• $\theta$ = angle between $\vec{p}$ and $\vec{E}$

Step 2 — Apply the Given Data

Given:

• $p = 4 \times 10^{-9} \, \text{C m}$
• $E = 5 \times 10^{4} \, \text{N C}^{-1}$
• $\theta = 30^\circ$

So:

Answer: $\boxed{1.0 \times 10^{-4} \, \text{N m}}$

Common Mistakes Students Make

✗ Mistake 1: Using $\cos\theta$ instead of $\sin\theta$

• Why it happens: Students confuse torque with the formula for potential energy ($U = -pE\cos\theta$).
• How to avoid:
  • Torque comes from the cross product → use $\sin\theta$.
  • Potential energy comes from the dot product → use $\cos\theta$.
  • Remember: Torque is maximum when dipole is perpendicular ($\theta = 90^\circ$) — that’s $\sin 90^\circ = 1$, not $\cos 90^\circ = 0$.

✗ Mistake 2: Taking $\theta$ as the angle with the field direction incorrectly

• Why it happens: Some problems give the angle between dipole and field as $60^\circ$ or $120^\circ$, and students use that directly without checking.
• How to avoid:
  • $\theta$ in $\tau = pE\sin\theta$ is always the angle between $\vec{p}$ and $\vec{E}$.
  • If the problem says “aligned at $30^\circ$ with the field”, that’s exactly $\theta = 30^\circ$ — correct here.

✗ Mistake 3: Forgetting to convert units or misreading powers of 10

• Why it happens: $p$ is given in $10^{-9}$ and $E$ in $10^{4}$ — students sometimes multiply without tracking exponents.
• How to avoid:
  • Write all numbers in scientific notation before multiplying.
  • Do exponent arithmetic separately: $10^{-9} \times 10^{4} = 10^{-5}$.

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