A 500 µC charge is at the centre of a square of side 10 cm. Find the work done in moving a charge of 10 µC between two diagonally opposite points on the square.

To find the work done in moving a charge of 10 µC between two diagonally opposite points on a square with a 500 µC charge at its center, we can use the concept of electric potential. The work done in moving a charge in an electric field is related to the difference in electric potential between the two points. Let's break this down step by step.

Understanding Electric Potential

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

V = k * (Q / r)

where:

• V is the electric potential in volts (V),
• k is Coulomb's constant, approximately 8.99 x 10^9 N m²/C²,
• Q is the charge creating the potential (in coulombs), and
• r is the distance from the charge to the point where the potential is being calculated (in meters).

Calculating the Distance

In this scenario, the charge of 500 µC is at the center of the square, and we need to find the distance from this charge to one of the corners of the square. The side length of the square is 10 cm, which is equivalent to 0.1 m. The distance from the center to a corner can be calculated using the Pythagorean theorem:

r = √((0.1/2)² + (0.1/2)²) = √(0.005² + 0.005²) = √(0.000025 + 0.000025) = √(0.00005) = 0.00707 m

Finding the Electric Potential at the Corners

Now, we can calculate the electric potential at one of the corners of the square:

V = k * (Q / r) = (8.99 x 10^9 N m²/C²) * (500 x 10^-6 C / 0.0707 m)

Calculating this gives:

V ≈ 8.99 x 10^9 * 7.07 x 10^-3 ≈ 63,700 V

Work Done in Moving the Charge

The work done (W) in moving a charge (q) between two points in an electric field is given by:

W = q * ΔV

Since the potential is the same at both corners (as they are equidistant from the charge at the center), the change in potential (ΔV) is zero:

ΔV = V1 - V2 = 63,700 V - 63,700 V = 0 V

Thus, the work done in moving the charge of 10 µC between the two diagonally opposite corners is:

W = 10 x 10^-6 C * 0 V = 0 J

Final Thoughts

In summary, since the electric potential is the same at both points, no work is done in moving the charge between them. This illustrates an important principle in electrostatics: when moving a charge within an equipotential surface, no work is required. If you have any further questions or need clarification on any part of this process, feel free to ask!