Problem 22 - Master Resource Book in JEE Main Physics

Question

A charged body has an electric flux \(\phi\) associated with it. The body is now placed inside a metallic container. The electric flux, \(\phi_{1}\) associated with the container will be (a) \(\phi_{1}=0\) (b) \(0<\phi_{1}<\phi\) (c) \(\phi_{1}=\phi\) (d) \(\phi_{1} \geq \phi\)

Step-by-Step Solution

Answer

(a) \( \phi_{1}=0 \) because the metallic container blocks the electric flux outside.

Step 1: Understanding Gauss's Law

According to Gauss's Law, the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of the medium. So, when a charge is placed inside a metallic container, the total flux is determined by the charge within the container.

Step 2: Charging Inside the Container

When the charged body is placed inside a metallic container, all the charge affects the electric field inside the metallic shell, but no field lines exit or penetrate the container to change the flux through its boundary.

Step 3: Applying Gauss's Law to the Container

Using Gauss's Law, the electric flux associated with the outer surface of the metallic container only depends on the charge enclosed. Since the container is metallic and fully surrounds the charge, it will neutralize any electric field within from leaving or entering.

Step 4: Conclusion with Gauss's Law

Based on Gauss's Law principles, since the metallic container blocks the electric field outside and no net flux can pass beyond, the associated flux with the container's surface becomes zero.

Key Concepts

Electric Flux

Electric flux is a key concept in electromagnetism, describing how much electric field passes through a given area. Mathematically, it is represented as \( \phi = \mathbf{E} \cdot \mathbf{A} \), where \( \mathbf{E} \) is the electric field and \( \mathbf{A} \) is the area vector. Essentially, it tells us how 'flowy' the electric field is through a surface.

Imagine an electric field being like a river and the surface like a net; the flux is the amount of river water flowing through the net. The stronger or denser the electric field, or the larger the area it affects, the greater the electric flux. It captures both the strength of the field and its direction relative to the surface.

• Positive flux occurs when field lines are exiting a closed surface.
• Negative flux occurs when field lines are entering a closed surface.

This concept is fundamental in understanding how electric fields interact with their environment and is pivotal in analyzing problems using Gauss's Law.

Charged Body

A charged body is simply an object that carries an excess of either positive or negative charge, typically measured in coulombs. These charges arise due to an imbalance between protons and electrons.

In the context of Gauss's Law, a charged body inside a closed surface can determine the net electric flux through that surface. The charge within dictates the field lines that originate or terminate at it. Understanding how charged bodies behave is crucial for predicting electric field interactions.

Key points to remember:

• If only a single charge is contained within a surface, that charge is solely responsible for the electric flux.
• The total charge within a boundary determines the net flux, regardless of charge distribution.

Utilizing these insights helps us unravel how charged objects affect their surrounding electric fields.

Metallic Container

A metallic container is unique in the realm of electric fields because metals are conductive materials. When a charged body is placed inside a metallic container, the conductors react to the electric charge.

Here's what happens:

• The electric field from the charged body causes the electrons in the metal to move.
• These electrons rearrange themselves on the container's inner surface.
• This rearrangement creates an opposing electric field inside the metal.

The result is that no net electric field exists within the metal itself. This phenomenon ensures the electric flux through the outer surface of the metallic container is effectively neutralized or zero.

The metal's conductive properties ensure the internal electric field effects are contained, preventing any external electric field from impacting the container.

Closed Surface

A closed surface is one that completely encloses a volume, separating the interior from the exterior. Examples of closed surfaces include spheres or cubes, with all points on the surface connected.

In electrical phenomena, closed surfaces are crucial because they allow us to apply Gauss's Law efficiently. The electric flux through a closed surface depends entirely on the charge enclosed within it, not on its shape or size.

• According to Gauss's Law, if there is no enclosed charge, the net flux is zero.
• Closed surfaces help in understanding field interactions, isolating internal effects from external ones.

These surfaces act as boundaries for imaginary field assessments, and when dealing with metallic enclosures, they confirm that all internal electric activity is canceled out, leading to zero flux beyond the exterior surface.