Problem 74 - Master Resource Book in JEE Main Physics

Question

A point charge \(q\) moves from point \(P\) to point \(S\) along the path \(P Q R S\) in a uniform electric field \(\mathbf{E}\) pointing parallel to the positive direction of the \(x\)-axis as shown in figure. The coordinates of the points \(P, Q, R \quad\) and \(S\) are \((a, b, 0)(2 a, 0,0),(a,-b, 0)\) and \((0,0,0)\) respectively. The work done by the field in the above process is given by the expression (a) \(q E\) (b) \(-q a E\) (c) \(q\left(\sqrt{a^{2}+b^{2}}\right)+E\) (d) \(3 q E\left(\sqrt{a^{2}+b^{2}}\right)\)

Step-by-Step Solution

Answer

The correct answer is (b) \(-q a E\).

Step 1: Understanding the Problem

We need to determine the work done by a uniform electric field on a point charge moving through a specified path between given coordinates. The problem involves understanding how work is calculated in an electric field.

Step 2: Recognizing Work Done by Electric Field Formula

The work done by an electric field on a charge moving from one point to another is given by the formula \( W = q \cdot (V_i - V_f) \), where \( V_i \) and \( V_f \) are the initial and final potentials. However, in terms of displacement along the direction of the field, work done can be placed directly using \( W = q \cdot E \cdot d \) where \( d \) is the component of displacement along the field direction.

Step 3: Determine Displacement Along Field Direction

Points can be described in terms of coordinates, where the field direction is intersected along the x-axis. The coordinates of P, Q, R, and S all differ in the x-axis direction. Compute the net change along the x-axis, \( x_s - x_p \), as this amounts to \(-a\) given that the x-component of the path starts at \(a\) and ends at \(0\).

Step 4: Calculate Work Done Using Displacement

Substitute the displacement into the work expression: \( W = q \cdot E \cdot (-a) \). This gives us \( W = -qEa \).

Step 5: Final Expression and Solution Selection

The calculated expression for work done is \(-qEa\). This matches with option (b) in the problem statement.

Key Concepts

Work Done

In physics, work done by a force is a measure of energy transfer when an object is moved over a distance by an external force applied in the direction of the movement. When it comes to electric fields, work done can specifically refer to the work done by or against an electric field as a point charge moves between two points.

The formula to calculate the work done by an electric field is given by:

• \( W = q \cdot (V_i - V_f) \)
• \( W = q \cdot E \cdot d \)

Here, \( q \) is the charge, \( E \) is the electric field strength, and \( d \) is the displacement in the field direction. The potential difference \( V \) can also be used when the displacement path is known along the electric lines.

The work done is a scalar quantity and may be positive or negative depending on whether the charge moves with or against the electric field lines. In this exercise, a point charge moves opposite to the electric field, resulting in negative work done, calculated as \(-qEa\).

Point Charge

A point charge is an idealized model used in physics to analyze the effect of electric fields in the simplest form. It assumes the electric charge is concentrated at a single point in space.

This concept allows us to simplify problems and focus on understanding the fundamental interactions between electric charges and fields without considering the dimensions of real objects.

For example, when analyzing the work done on a point charge moving in an electric field, we can directly apply the respective formulas without worrying about the physical size or shape of the charge. This is crucial to solving problems, as seen in our problem where we used the concept of a point charge to derive the work done expression as \( W = -qEa \).

Displacement in Electric Field

Displacement in an electric field refers to the movement of a charge within that field. It is essential to understand how displacement affects work done in an electric field.

The component of displacement along the field direction is crucial because only this part contributes to the work done by the field. This is why the formula \( W = q \cdot E \cdot d \) uses the displacement along the direction of the electric field.

In our exercise, we calculated the displacement by determining the difference in coordinates for the point charge along the x-axis, which is aligned with the electric field. The net change was found to be \(-a\), given the coordinates moved from \(a\) to \(0\). Understanding this directional displacement helps derive the correct expression for work done in electric fields.