Torque Acting on a Dipole: Comprehensive NEET Physics Notes

1. Torque on an Electric Dipole in a Uniform Electric Field

1.1 Introduction to Torque on a Dipole

An electric dipole consists of two equal and opposite charges, separated by a small distance. When this dipole is placed in a uniform electric field, each charge experiences a force, but these forces are equal and opposite. This results in a net torque acting on the dipole, causing it to rotate rather than move in a straight line.

The electric dipole moment is represented by the vector $\mathbf{p}$, which has a magnitude given by $p=q\times 2a$ (where $q$ is the charge and $2a$ is the separation between the charges). The direction of the dipole moment is from the negative to the positive charge.

The torque $\tau$ acting on a dipole placed in a uniform electric field $\mathbf{E}$ is given by:

$\tau =\mathbf{p}\times \mathbf{E}$

The magnitude of the torque is:

$\tau =pE\sin \theta$

Where:

• $\theta$ is the angle between the dipole moment vector $\mathbf{p}$ and the electric field vector $\mathbf{E}$.
• $p$ is the magnitude of the dipole moment.
• $E$ is the magnitude of the electric field.

Did You Know?

• Electric dipoles naturally align themselves with external electric fields. This alignment minimizes the potential energy, which is why torque acts on the dipole to rotate it.

1.2 Torque and Dipole Alignment

If the dipole is aligned with the electric field ($\theta =0^{\circ }$ or $\theta =180^{\circ }$), the torque is zero. This happens because the forces on the charges act along the same line, and there is no tendency for the dipole to rotate. The torque is maximum when the dipole is perpendicular to the electric field ($\theta =90^{\circ }$).

At equilibrium, the dipole tends to align itself in the direction of the electric field, minimizing its potential energy.

1.3 Work Done by the Torque

The work done by the torque when rotating a dipole from an initial angle ${\theta }_1$to a final angle ${\theta }_2$​ is:

$W=pE(\cos {\theta }_1-\cos {\theta }_2)$

This work is stored as potential energy. The potential energy UUU of a dipole in a uniform electric field at an angle θ\thetaθ is:

$U=-pE\cos \theta$

Real-life Application:

• The alignment of water molecules (which are dipoles) in electric fields plays a crucial role in biological processes, such as the behavior of cells in living organisms and the function of chemical reactions in aqueous solutions.

Quick Recap:

• Torque on a dipole: $\tau =\mathbf{p}\times \mathbf{E}$
• Magnitude of torque: $\tau =pE\sin \theta$
• Potential energy: $U=-pE\cos \theta$
• Work done by torque: $W=pE(\cos {\theta }_1-\cos {\theta }_2)$

NEET Tip:

• Remember that torque is maximized when the dipole is perpendicular to the electric field ($\theta =90^{\circ }$). This scenario is often tested in NEET problems involving rotational equilibrium.

2. Visualizing Torque with Diagrams

To help you understand the concept of torque more clearly, consider the following diagram suggestions:

• Dipole in Electric Field: A diagram showing a dipole in a uniform electric field with forces acting on each charge, and the resulting torque causing rotation.
• Vector Diagram for Torque: A depiction of the vector cross-product of $\mathbf{p}$ and $\mathbf{E}$, showing how torque is calculated.

2.1 Supplementary Visual Aids

Including diagrams of a rotating dipole in an electric field and vector representations of the torque will greatly aid in visualizing how the dipole interacts with external forces. These diagrams will highlight key points where torque is zero (aligned with the field) and where it is maximized (perpendicular to the field).

3. NEET-Specific Practice Questions

Question 1:

An electric dipole has a dipole moment of $4\times 10^{-6}\text{Cm}$ and is placed in a uniform electric field of $10^4\text{N/C}$. Calculate the torque acting on the dipole when the angle between the dipole moment and the electric field is $45^{\circ }$.

Solution: Given:

• $p=4\times 10^{-6}\text{Cm}$
• $E=10^4\text{N/C}$
• $\theta =45^{\circ }$

The torque is: $\tau =pE\sin \theta$ $\tau =(4\times 10^{-6})\times (10^4)\times \sin 45^{\circ }$ $\tau =4\times 10^{-2}\times \frac{1}{\sqrt{2}}$$\tau \approx 2.83\times 10^{-2}\text{Nm}$

Question 2:

A dipole with a moment of $5\times 10^{-5}\text{Cm}$ is placed in an electric field of $2\times 10^3\text{N/C}$. Calculate the potential energy when the dipole is at an angle of 60∘60^\circ60∘ with the field.

Solution: Given:

• $p=5\times 10^{-5}\text{Cm}$
• $E=2\times 10^3\text{N/C}$
• $\theta =60^{\circ }$

The potential energy is: $U=-pE\cos \theta$ $U=-(5\times 10^{-5})\times (2\times 10^3)\times \cos 60^{\circ }$ $U=-(5\times 10^{-5})\times (2\times 10^3)\times \frac{1}{2}$ $U=-5\times 10^{-2}\text{J}$

Question 3:

If a dipole with a moment of $3\times 10^{-8}\text{Cm}$ is aligned parallel to a uniform electric field of $1.5\times 10^5\text{N/C}$, what is the torque acting on it?

Solution: Since the dipole is parallel to the electric field ($\theta =0^{\circ }$), the torque is:τ=pEsin0<