Problem 65 - Master Resource Book in JEE Main Physics

Question

A cylinder of radius, \(R\) and length, \(L\) is placed in a uniform electric field, \(E\) parallel to the cylinder axis. The total flux for the surface of the cylinder is given by (a) zero (b) \(\pi R^{2} / E\) (c) \(2 \pi R^{2} E\) (d) None of these

Step-by-Step Solution

Answer

The total electric flux is zero, answer is (a).

Step 1: Understand the Problem

We need to determine the electric flux through the surface of a cylinder when a uniform electric field is parallel to the axis of the cylinder. The possible answers are (a) zero, (b) \(\pi R^{2} / E\), (c) \(2 \pi R^{2} E\), or (d) none of these.

Step 2: Recall Gauss's Law

According to Gauss's Law, the total electric flux \(\Phi\) through a closed surface is equal to the charge enclosed divided by the permittivity of free space, i.e., \(\Phi = \frac{Q_{enc}}{\varepsilon_0}\). Here, we have no charge enclosed within the surface of the cylinder since the electric field is parallel to the axis and there is no charge on the cylinder.

Step 3: Calculate Flux For Each Cylinder Part

The cylinder is composed of a curved surface and two flat circular ends. The electric field is parallel to the cylinder, so it will not penetrate the curved surface at all, producing zero flux there. On the two circular ends, the flux is calculated as \(E \times A\), where \(A\) is the area of the circle, but the contribution from one end cancels the other because the electric field enters one end and leaves the other.

Step 4: Find Net Electric Flux

Since the electric flux from the curved surface is zero and the contributions from the circular faces cancel each other, the net electric flux through the entire closed surface is zero.

Step 5: Conclusion

Therefore, based on the steps above and the application of Gauss's law, the total electric flux through the closed surface of the cylinder is zero, which matches option (a).

Key Concepts

Electric Flux

Electric flux is a fundamental concept in electromagnetism that describes how an electric field interacts with a given surface. The electric flux through a surface can be thought of as the number of electric field lines passing perpendicularly through that surface. Mathematically, it's expressed as:\[\Phi = \int \vec{E} \cdot \vec{A}\]where \( \Phi \) is the electric flux, \( \vec{E} \) is the electric field vector, and \( \vec{A} \) is the area vector of the surface. The dot product \( \vec{E} \cdot \vec{A} \) signifies that only the component of the electric field perpendicular to the surface contributes to the flux. This makes the concept of electric flux intuitive: if a surface is aligned such that an electric field flows directly through it, maximum flux is achieved.

• Positive flux: when field lines exit the surface.
• Negative flux: when field lines enter the surface.

Understanding how electric flux works through surfaces can give deep insights into how fields act in a physical space, aiding in practical applications like electric field analysis.

Closed Surface

A closed surface is one that completely encloses a volume without any breaks or gaps. Think of it as a surface that forms a 'shell' around a space, like a balloon or a soap bubble. In Gauss's Law, evaluating electric flux through closed surfaces is crucial, since it allows for the deduction of enclosed charge within the surface. In the context of the problem, the cylinder is a closed surface. It includes not only the sides but also the flat circular caps at each end. This makes it important to consider all different parts of this structure when calculating flux.

• The curved surface of a cylinder contributes zero to net flux when the field is parallel to the axis.
• Circular ends need careful consideration since they can have contributions that may cancel out.

With Gauss's Law stating that total flux through a closed surface equals the charge enclosed divided by permittivity, analyzing closed surfaces is key, even when no charge is enclosed, as it simplifies understanding fields and flux behavior.

Uniform Electric Field

A uniform electric field is characterized by field lines that run parallel to each other, maintaining the same field strength and direction across the space. This uniformity makes calculations involving such fields simpler and more predictable compared to non-uniform fields.In the problem context, the electric field \(E\) is uniform and aligned parallel to the cylinder's axis. This means that the field's influence is evenly distributed and more predictable.

• No penetration through the curved surface: flux through curved surfaces of angled objects are zero in a parallel field.
• Consideration of net positive and negative contributions on opposite ends.

A uniform field simplifies understanding and solving problems because the constant nature of the field's strength and direction ensures that calculations, like those for electric flux, are straightforward and systematic.