Two charges 5 × 10–8 C and –3 × 10–8 C are located 16 cm apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero

To determine the points along the line joining two charges where the electric potential is zero, we need to consider the contributions of both charges to the electric potential at various points. The charges in question are +5 × 10⁻⁸ C and -3 × 10⁻⁸ C, separated by a distance of 16 cm. Let's break this down step by step.

Understanding Electric Potential

The electric potential (V) at a point in space due to a point charge is given by the formula:

V = k * Q / r

where:

• k is Coulomb’s constant (approximately 8.99 × 10⁹ N m²/C²),
• Q is the charge, and
• r is the distance from the charge to the point where the potential is being calculated.

The total electric potential at any point due to multiple charges is the algebraic sum of the potentials due to each charge.

Setting Up the Problem

Let’s denote the position of the positive charge (+5 × 10⁻⁸ C) as point A and the negative charge (-3 × 10⁻⁸ C) as point B. The distance between these charges is 16 cm, which we can convert to meters for our calculations (0.16 m).

We are interested in finding points along the line connecting A and B where the total potential V is zero:

V_total = V_A + V_B = 0

Finding the Points of Zero Potential

We will consider two regions for potential zero: one region to the left of charge A and another to the right of charge B.

Region 1: To the Left of Charge A

Let’s denote a point P located at a distance x from charge A (to the left). The distances from the charges to point P will be:

• Distance from A to P: r_A = x
• Distance from B to P: r_B = (0.16 + x)

The electric potential at point P is:

V_P = k * (5 × 10⁻⁸) / x - k * (3 × 10⁻⁸) / (0.16 + x)

Setting this equal to zero gives us:

5 × 10⁻⁸ / x = 3 × 10⁻⁸ / (0.16 + x)

Cross-multiplying yields:

5(0.16 + x) = 3x

Simplifying this equation:

0.8 + 5x = 3x

2x = -0.8

x = -0.4 m

This means there is a point 40 cm to the left of charge A where the potential is zero.

Region 2: To the Right of Charge B

Now, let’s find the point P on the right side of charge B, at a distance y from charge B. The distances from the charges will be:

• Distance from A to P: r_A = (0.16 + y)
• Distance from B to P: r_B = y

The potential is:

V_P = k * (5 × 10⁻⁸) / (0.16 + y) - k * (3 × 10⁻⁸) / y

Setting this equal to zero:

5 × 10⁻⁸ / (0.16 + y) = 3 × 10⁻⁸ / y

Cross-multiplying gives:

5y = 3(0.16 + y)

Simplifying:

5y = 0.48 + 3y

2y = 0.48

y = 0.24 m

This means there is also a point 24 cm to the right of charge B where the potential is zero.

Final Summary

The points along the line joining the two charges where the electric potential is zero are:

• 40 cm to the left of charge A
• 24 cm to the right of charge B

These calculations demonstrate how to find points of zero potential in the electric field created by point charges. Understanding these principles is crucial for mastering concepts in electrostatics.