Problem 61 - Master Resource Book in JEE Main Physics

Question

Two equal charges \(q\) of opposite sign separated by a distance \(2 a\) constitute an electric dipole of dipole moment \(p\). If \(P\) is a point at a distance \(r\) from the centre of the dipole and the line joining the centre of the dipole to this point makes and angle \(\theta\) with the axis of the dipole, then the potential at \(P\) is given by \((r>>2 a)\) (where, \(p=2 q a)\) (a) \(V=\frac{p \cos \theta}{2 \pi \varepsilon_{0} r^{2}}\) (b) \(V=\frac{p \sin \theta}{4 \pi \varepsilon_{0} r}\) (c) \(V=\frac{p \cos \theta}{4 \pi \varepsilon_{0} r}\) (d) \(V=\frac{p \cos \theta}{4 \pi \varepsilon_{0} r^{2}}\)

Step-by-Step Solution

Answer

(d) \(V=\frac{p \cos \theta}{4 \pi \varepsilon_{0} r^{2}}\)

Step 1: Understand the Dipole Moment

The dipole moment \( p \) of an electric dipole is given by \( p = 2qa \), where \( q \) is the charge and \( a \) is the half-distance between the charges. This setup forms the basis for calculating the potential at point \( P \).

Step 2: Use the Formula for Electric Potential of a Dipole

The electric potential \( V \) at a point \( P \), which is at a distance \( r \) from the center of an electric dipole, is given by the standard formula: \[ V = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2}, \]where \( \theta \) is the angle between the dipole axis and the line joining the dipole center to point \( P \). This relation holds when \( r \gg 2a \), indicating that the point is far from the dipole compared to its size.

Step 3: Match the Formula with Given Options

Compare the formula obtained, \[ V = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2}, \] with the options provided. The option that matches is (d) \( V = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2} \).

Key Concepts

Electric Dipole

An electric dipole is a simple system consisting of two equal but opposite charges that are closely spaced. Imagine them as two little charges, one positive and one negative, always sticking close to each other like a pair of magnets. These charges create a special kind of electric field because they distort the space around them in a unique way.
Understanding what an electric dipole is can help you see how these charges interact with electric fields and other charges:

• It is made up of two charges, +q and -q.
• These charges are separated by a distance of 2a.
• The entire system is electrically neutral because the positive and negative charges cancel each other out.

These properties make the electric dipole an essential topic in both physics and engineering, especially when studying molecules and their interactions.

Dipole Moment

The dipole moment is a fundamental concept used to describe the strength and direction of an electric dipole. Essentially, it is a measure of how much torque the dipole experiences in an electric field, or how strong the dipole is.
The formula for the dipole moment \( p \) is given by \( p = 2qa \), where:

• \( q \) is the magnitude of one of the charges.
• \( a \) is the distance from the center to one of the charges (half the distance between the charges).

This formula indicates that both the amount of charge and the separation between the charges are crucial in determining the dipole moment.
A larger dipole moment means the dipole has a stronger ability to align with an electric field. It's like having a more powerful magnet, pulling itself to line up with the field more effectively. Remember, dipoles in molecules can affect how they interact with each other, impacting everything from state changes to chemical reactions.

Electric Potential

The electric potential due to an electric dipole is a measure of the potential energy a charge would have at a given point in space. This potential is a little different from simple point charges because a dipole has two charges with opposite signs.
For a point \( P \) at a distance \( r \) from the center of the dipole and an angle \( \theta \) with respect to the dipole axis, the electric potential \( V \) is calculated using a specific formula: \[ V = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2} \]Where:

• \( p \) is the dipole moment, \( 2qa \).
• \( \theta \) is the angle with the dipole axis.
• \( \varepsilon_0 \) is the permittivity of free space.

This formula is relevant when \( r \) is much larger than the size of the dipole, meaning the point is quite far away.
This electric potential helps us understand how electric fields affect charges at different positions relative to dipoles. It's a crucial aspect for things like capacitance in capacitors and even how cellular membranes work in biology.