Problem 9 - Master Resource Book in JEE Main Physics

Question

Work done in carrying a charge \(Q_{1}\) once round a circle of radius \(R\) with a charge \(Q_{2}\) at the centre is (a) \(\frac{Q_{1} Q_{2}}{4 \pi \varepsilon_{0} R^{2}}\) (b) zero (c) \(\frac{Q_{1} Q_{2}}{4 \pi \varepsilon_{0} R}\) (d) infinite

Step-by-Step Solution

Answer

The work done is zero.

Step 1: Understanding the Problem

We are asked to find the work done in carrying a charge \( Q_1 \) around a circle of radius \( R \) with a charge \( Q_2 \) at the center. Options are given which suggest a calculation involving charges, distance, or potentially zero, so we need to consider the properties of electrostatic fields.

Step 2: Recall the Concept of Work in Electric Fields

In electrostatics, the work done by or against the electric field when moving a charge along a path is given by the difference in electric potential energy. For a closed path, the work done is the integral of the electric field \( \mathbf{E} \) dot product with the path element \( d\mathbf{s} \).

Step 3: Applying Gauss' Law

For a charge \( Q_2 \) at the center of a circle (or sphere), the electric field at any point on the circle's circumference is radial. According to Gauss' Law, the electric flux through a closed surface surrounding the charge is \( \frac{Q_2}{\varepsilon_0} \). However, along a circular path, \( abla \times \mathbf{E} = 0 \) in static fields.

Step 4: Calculate Work for Moving Charge in a Circle

Because the electric field is conservative, the work done moving charge \( Q_1 \) in a complete circle around \( Q_2 \) is zero. This is because potential difference between any point and itself is zero, thus no net work is done.

Step 5: Conclusion

Since the charge completes a loop, the potential energy difference is zero. Hence, the work done is zero.

Key Concepts

Electric Potential Energy

Electric potential energy is the energy stored due to the positions of charges within an electric field. Imagine it as the potential energy you store when you lift an object against gravity. Here, electrical forces are at play instead of gravitational forces.

When a charge, say charge \( Q_1 \), is placed in the electric field of another charge, \( Q_2 \), it experiences a force. This means that moving the charge \( Q_1 \) within the field, especially closer or further from \( Q_2 \), would require doing some electric work. The work done changes the potential energy within the system.

If you picture the scenario where \( Q_1 \) is made to move in a circle around \( Q_2 \) and returns to its starting location, something interesting happens. Since there's no net change in distance between the charges in one complete roundtrip, the electric potential energy remains constant. Thus, the energy input needed to travel the circular path isn't cumulative. The total work done remains zero, much like walking in a circle and ending where you started.

Gauss's Law

Gauss's Law is a fundamental principle in electrostatics, providing a relation between electric fields and the charges that cause them. It states that the total electric flux passing through any closed surface is proportional to the enclosed charge.

The law's mathematical expression is \( \Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0} \), where \( \Phi_E \) is the electric flux, \( \mathbf{E} \) is the electric field, and \( Q_{enc} \) is the enclosed charge. \( \varepsilon_0 \) is the permittivity of free space.

In our problem, with a charge \( Q_2 \) at the center of the circle, the electric field at any point on the circle's surface is radial. For a symmetrical arrangement, like a circle around a central point charge, the electric flux through the surface does not change, and neither does the radial field along our path. This understanding helps us conclude that the field doesn't exert any net work as \( Q_1 \) moves along its closed circular path.

Conservative Electric Fields

Conservative electric fields are fields where the work done in moving a charge between two points is independent of the path taken. It's much like walking two different paths on a hill and ending up with the same elevation change, thus exerting the same amount of effort.

For such fields, we define a scalar electric potential that depends only on the position, not the path. The key characteristic is that any closed loop, such as our charge path, results in zero net work.

• Electric fields created by static charges are conservative.
• In closed paths, like a circle around a charge, these fields ensure the potential difference between starting and ending points remains zero.
• Thus, reinforcing why moving \( Q_1 \) once around \( Q_2 \) results in zero work done.

This intrinsic property makes them excellent for energy conservation analyses in electrostatic problems, helping us realize, as in the exercise, that no extra work is needed to bring things back to how they began.