Question

A ring of radius \(R\) is having two charges \(q\) and \(2 q\) distributed on its two half parts. The electric potential at a point on its axis at a distance \(2 \sqrt{2} \mathrm{R}\) from its centre is : (1) \(\frac{3 \mathrm{kq}}{\mathrm{R}}\) (2) \(\frac{\mathrm{kq}}{3 \mathrm{R}}\) (3) \(\frac{\mathrm{kq}}{\mathrm{R}}\) (4) \(\frac{\mathrm{kq}}{\sqrt{3 \mathrm{R}}}\)

Short answer

kq / R.

Step-by-step solution

Understanding the problem

A ring of radius R has two charges, q and 2q, distributed over its two halves. We need to find the electric potential at a point on its axis at a distance of 2√2R from its center.

Recall the formula for electric potential

The electric potential due to a ring of charge at a point on its axis is given by: V = kQ/√(R² + z²) where Q is the total charge on the ring, R is the radius of the ring, and z is the distance from the center of the ring to the point.

Calculate the electric potential for each half

For the half with charge q: V₁ = kq / √(R² + (2√2R)²) For the half with charge 2q: V₂ = k(2q) / √(R² + (2√2R)²)

Simplify the expressions

Combine the charges using the given distance: V₁ = kq / √(R² + 8R²) = kq / (3R)V₂ = k(2q) / √(R² + 8R²) = k(2q) / (3R)

Add the potentials

Add the potentials from both halves: V_total = V₁ + V₂ = (kq / 3R) + (2kq / 3R) = 3kq / 3R = kq / R

Conclusion

The electric potential at the given point is kq / R.

Key concepts

Electrostatics

Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from charged particles that are at rest. In electrostatic systems, charges build up on surfaces and contribute to electric fields and potentials.

When analyzing a problem in electrostatics, we focus on static charges. We understand how charges distribute themselves and how they interact by creating forces on each other. In this exercise, we examine a ring with varying charge distributions on its halves, and we look at the electric potential on the ring's axis.

Electrostatic principles help predict the electric potential due to a distribution of charges, such as in this problem where the charges are symmetrically placed on a ring. By solving the problem using these principles, we can determine the potential at a specific point, helping us understand how electric potentials combine and influence points in space.

Electric Potential

Electric potential is the amount of electric potential energy per unit charge at a specific point in an electric field. It is a scalar quantity and is often denoted by the symbol V. Mathematically, the electric potential V at a point due to a charge Q at a distance r is given by the formula:

\( V = \frac{kQ}{r} \)

Here, k is Coulomb's constant (\( k = 8.99 \times 10^9 \) Nm²/C²).

In this exercise, the electric potential is calculated at a point on the axis of a charged ring. For each half of the ring with charge q and 2q, the potential is:
\( V₁ = \frac{kq}{\text{√}(R² + (2\text{√}2R)²)} \) and \( V₂ = \frac{k(2q)}{\text{√}(R² + (2\text{√}2R)²)} \) respectively.

We simplify these expressions by noting that \( \text{√}(R² + 8R²) = 3R \), leading to: \( V₁ = \frac{kq}{3R} \) and \( V₂ = \frac{2kq}{3R} \).

Adding these potentials, the total electric potential at the given point is \( V_{\text{total}} = \frac{kq}{3R} + \frac{2kq}{3R} = \frac{3kq}{3R} = \frac{kq}{R} \). Thus, the electric potential at the point 2√2R from the center of the ring is \( V = \frac{kq}{R} \).

Charge Distribution

Charge distribution refers to how electric charge is distributed over a physical body or region in space. The uniform or non-uniform distribution of charge can significantly affect the electric field and potential in the region around it.

In this exercise, we have a ring with two different charges, q and 2q, distributed over its two halves. One half has a uniform charge of q, and the other half has a charge double the first (2q).

Charge distribution on the ring creates a unique electric potential at various points along the ring's axis. To find the potential at a point on the axis, we consider how each segment of the ring contributes to the overall potential.

Using the formula for the potential due to a ring of charge, we sum contributions from each half. Each segment's potential is then added to find the total electric potential. This exercise illustrates how different charge distributions result in electric potentials that can be combined to determine the field influence at a point.

By understanding charge distribution and its impact on electric potential, we can solve more complex problems and gain deeper insight into electrostatics and its applications.