Electric Field and Potential Due to Two Point Charges: A Comprehensive Analysis

Introduction

Understanding the electric field and potential due to charged particles is crucial in physics and engineering applications. In this article, we explore a specific scenario involving two point charges, focusing on the points along the x-axis where the electric field and electric potential are zero.

Problem Setup

Consider two point charges, q1 and q2, placed along the x-axis. Here, q1 3.0 μC and q2 -2.0 μC, with a distance of 4.0 cm between them. q1 is located at the origin (0, 0) and q2 is placed at (0.04, 0).

Step-by-Step Solution

Electric Field (E)

The electric field due to a point charge is given by:

E k · q / r

where k is Coulomb's constant, q is the charge, and r is the distance from the charge.

Left of q1

To the left of q1 (x 0), both fields are in the same direction (to the right). Therefore, the electric field cannot be zero.

Between the Charges (0 x 0.04)

Let x be the distance from q1 to a point where the electric field is zero. The distances from the charges to this point are r1 x and r2 0.04 - x.

Setting the magnitudes of the electric fields equal:

E1 E2 rarr; k · 3.0 × 10^(-6) / x^2 k · 2.0 × 10^(-6) / (0.04 - x^2)

Canceling out k:

3.0 × 10^(-6) / x^2 2.0 × 10^(-6) / (0.04 - x^2)

30.04 - x^2 2x^2

30.0016 - 0.08x 3x^2

0.0016 - 0.24x 3x^2 0

Using the quadratic formula:

x (-b ± sqrt{b^2 - 4ac}) / 2a (0.24 ± sqrt{0.24^2 - 4 · 1 · 0.0048}) / 2

(0.24 ± sqrt{0.0576 - 0.0192}) / 2 (0.24 ± sqrt{0.0384}) / 2

(0.24 ± 0.196) / 2

Calculating the two possible values for x:

x (0.436) / 2 0.218 m (not valid, as it is outside the range) x (0.044) / 2 0.022 m (valid)

Right of q2

To the right of q2 (x 0.04), both electric fields point to the left. Therefore, the electric field cannot be zero.

Electric Potential (V)

The electric potential due to a point charge is given by:

V k · q / r

The total potential at a point x is:

Vtotal V1 V2 k · 3.0 × 10^(-6) / x k · -2.0 × 10^(-6) / (0.04 - x)

Setting Vtotal 0:

(3.0 / x) (2.0 / (0.04 - x))

Cross-multiplying:

3.0 · (0.04 - x) 2.0 · x

0.12 - 3. 2.

0.12 5.

x 0.12 / 5 0.024 m (2.4 cm from q1)

Summary

Electric Field is zero at:

i x 0.022 m (2.2 cm from q1)

Electric Potential is zero at:

i x 0.024 m (2.4 cm from q1)

Conclusion

The calculations show that the electric field and potential are zero at specific points along the x-axis. These results are significant for understanding complex charge distributions and their effects on the surrounding space.