Problem 156 - Master Resource Book in JEE Main Physics

Question

A uniformly charged thin spherical shell of radius \(R\) carries uniform surface charge density of \(\sigma\) per unit area. It is made of two hemispherical shells, held together by pressing them with force \(F\) (see figure) then \(F\) is proportional to [IIT JEE 2010] (a) \(\frac{1}{\varepsilon_{0}} \sigma^{2} R^{2}\) (b) \(\frac{1}{\varepsilon_{0}} \sigma^{2} R\) (c) \(\frac{1}{\varepsilon_{0}} \frac{\sigma^{2}}{R}\) (d) \(\frac{1}{\varepsilon_{0}} \frac{\sigma^{2}}{R^{2}}\)

Step-by-Step Solution

Answer

Option (a) \( \frac{1}{\varepsilon_{0}} \sigma^2 R^2 \).

Step 1: Understand the Configuration

The spherical shell is made of two hemispherical shells, which means there is an internal electrostatic pressure due to repulsive forces between the charges on these hemispheres. The charge density is defined as \( \sigma \), and \( R \) is the radius of the shell.

Step 2: Expression for Electrostatic Pressure

The electrostatic pressure \( P \) on a charged surface can be calculated using the formula: \( P = \frac{\sigma^2}{2 \varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space.

Step 3: Calculate the Force Required

The force \( F \) required to hold the hemispheres together is due to the electrostatic pressure over the curved surface area of one hemisphere. The area of a hemisphere is \( 2\pi R^2 \). Thus, the force is \( F = P \times 2\pi R^2 = \frac{\sigma^2}{2 \varepsilon_0} \times 2\pi R^2 = \frac{\pi \sigma^2 R^2}{\varepsilon_0} \).

Step 4: Identify the Proportionality

The force \( F \) is directly proportional to \( \frac{1}{\varepsilon_0} \sigma^2 R^2 \). Refer to the options to identify the correct proportional expression, which is option (a).

Key Concepts

Electrostatic Pressure

Electrostatic pressure is a fundamental concept when dealing with charged surfaces in physics. Imagine it like a tiny pushing force that occurs on charged surfaces due to repulsive electrical forces. When a surface holds a uniform charge density, all like-charges on the surface repel each other, resulting in this pressure.

By definition, electrostatic pressure is defined as the pressure exerted by a uniformly charged surface. Mathematically, it is given by the formula:

• \( P = \frac{\sigma^2}{2 \varepsilon_0} \)

where:

• \( \sigma \) is the surface charge density, representing the amount of charge per unit area.
• \( \varepsilon_0 \) is the permittivity of free space, a constant value describing how electric field lines propagate in a vacuum.

Significantly, this pressure becomes crucial when understanding forces within charged surfaces, especially when these surfaces are attempting to separate due to these internal electrostatic forces.

Permittivity of Free Space

The permittivity of free space, denoted by \( \varepsilon_0 \), is a key constant in electromagnetism. It describes the ability of a vacuum to allow electric field lines to pass through it. This constant is important when calculating electric fields and forces in vacuum conditions.

In terms of physics, \( \varepsilon_0 \) has implications in various equations related to electrostatics. It primarily features in Coulomb's Law, which governs the force between point charges in a vacuum:

• \( F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \)

Here, \( \varepsilon_0 \) helps quantify how much the electric forces between the charges are reduced compared to if the charges were situated in a free, empty space. In relation to electrostatic pressure, it serves as a denominator, showing that the amount of pressure exerted is inversely proportional to the permittivity of the medium.

Surface Charge Density

Surface charge density \( \sigma \) is a measure of how much electric charge is present per unit area on a surface. In simpler terms, it tells us how densely packed the charges are on a given surface.

This concept is crucial when analyzing electric fields generated by surfaces rather than point charges. For a uniform surface charge density, every part of the surface holds the same amount of charge, leading to consistent electric field distribution.

Mathematically, surface charge density is calculated as:

• \( \sigma = \frac{Q}{A} \)

where:

• \( Q \) is the total charge on the surface.
• \( A \) is the area of the surface.

In scenarios involving hemispherical surfaces or spherical shells, \( \sigma \) plays a vital role in determining the electrostatic pressure and the corresponding force required to keep these charged bodies intact, as all properties are directly influenced by how the charges are spread over the surface.