Q6. A thin plastic rod has length and a nonuniform linear charge density , where . With at infinity, find the electric potential at point on the axis, at a distance from one end.

Background

Topic: Electric Potential from Continuous Charge Distributions

This question tests your understanding of how to calculate the electric potential at a point due to a nonuniformly charged rod using integration. The rod's charge density varies with position, so you must set up and evaluate an integral for the potential.

Key Terms and Formulas

• Electric Potential (): The work done per unit charge in bringing a test charge from infinity to a point in space.

• Linear Charge Density (): Charge per unit length, here given as .

• Potential due to a Point Charge:

• Potential due to a Continuous Distribution:

• Permittivity of Free Space ():

Step-by-Step Guidance

1. Set up a coordinate system with the rod along the -axis, starting at and ending at . Point is located on the $x$-axis at a distance from the left end of the rod.

2. Consider an infinitesimal segment of the rod at position with length . The charge on this segment is .

3. The distance from this segment to point is (since $P_1$ is units to the left of the rod's start).

4. Write the expression for the infinitesimal potential at due to :

5. Set up the integral for the total potential at by integrating from $0L$:

Try solving on your own before revealing the answer!