Problem 77 - Fundamentals of Physics

Question

A spherical conducting shell has a charge of \(-14 \mu \mathrm{C}\) on its outer surface and a charged particle in its hollow. If the net charge on the shell is \(-10 \mu \mathrm{C},\) what is the charge (a) on the inner surface of the shell and (b) of the particle?

Step-by-Step Solution

Answer

(a) +4 μC; (b) -4 μC for the particle.

Step 1: Understand the problem

The problem involves a spherical conducting shell with a net charge and charges on both its outer surface and inside a hollow. We aim to find the charges on the inner surface of the shell and the charge of the particle inside the hollow.

Step 2: Use Gauss' Law Concept

According to electrostatic principles (like Gauss' Law), the charge enclosed by a conductor is distributed entirely on the outer surface, resulting in no electric field within the conductor's material.

Step 3: Calculate total charge on the shell

The net charge on the shell is given as \(-10 \mu \mathrm{C}\). Having \(-14 \mu \mathrm{C}\) on the outer surface means the charge on the inner surface must account for the net charge of the shell.

Step 4: Find the charge on the inner surface

The charge on the outer surface is \(-14 \mu \mathrm{C}\). Since the net charge is \(-10 \mu \mathrm{C}\), the inner surface must have a charge of \(+4 \mu \mathrm{C}\) to make the total charge \(-10 \mu \mathrm{C}\).

Step 5: Calculate the charge of the particle

The inner surface charge matches the negative of the enclosed particle's charge. Therefore, the charge on the particle must be \(-4 \mu \mathrm{C}\) to balance it and bring the inner surface charge to zero when considering inner shell equilibrium.

Key Concepts

Gauss' Law

Gauss' Law is a fundamental principle in electromagnetism, crucial for understanding electric fields in and around charged objects. It states that the total electric flux through a closed surface is equal to the charge enclosed by that surface divided by the electric constant (also known as the permittivity of free space). This can be mathematically represented as: \[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} \]where

• \( \Phi_E \) is the electric flux,
• \( Q_{enc} \) is the total charge enclosed within the surface,
• \( \varepsilon_0 \) is the permittivity of free space.

For a conductor in electrostatic equilibrium, the electric field inside is zero, and any excess charge resides on the surface. In this exercise, Gauss' Law helps us determine how charges are distributed on the inner and outer surfaces of the spherical shell.

net charge

The net charge of a system is the sum of all individual charges present within the system. For the spherical conducting shell, the net charge is given as \(-10 \mu \mathrm{C}\). However,

• the outer surface charge is given as \(-14 \mu \mathrm{C}\),
• indicating some charge on the inner surface to account for the total net charge.

By knowing the net charge, we can deduce other unknown contributions, such as how much charge must lie on the inner surface to satisfy this net value. In this case, since the outer charge is negative, the inner surface must have a positive charge to satisfy the overall charge of the shell.

inner surface charge

Understanding the charge on the inner surface of a conducting shell is crucial when dealing with a net charge that differs from the outer surface. Due to the principles of electrostatics, the inner surface of a conducting shell acquires a charge that precisely compensates for other enclosed charges,

• ensuring that the conductor's overall net charge is maintained.

Given our shell's known \(-14 \mu \mathrm{C}\) on the outer surface with a net charge of \(-10 \mu \mathrm{C}\),we calculate the inner surface must carry a charge of \(+4 \mu \mathrm{C}\).
This positive charge balances the computations, ensuring that the charges across the surfaces equate to the net charge given by the problem.

hollow charged particle

A hollow charged particle inside a conducting shell presents an interesting scenario due to the influence of surrounding electrostatic fields. The conducting nature of the shell ensures that a charge within the cavity doesn't affect the external surface charge. However, it has a direct impact on the charges on the inner surface.
In this problem, the charge of the hollow particle is

• \(-4 \mu \mathrm{C}\),

which precisely counterbalances the inner surface charge of \(+4 \mu \mathrm{C}\),leading to an electrostatic equilibrium inside the shell.
This balance is consistent with Gauss' Law, where the electric field inside the conductive material and shell remains unaffected by internal charges, ensuring the shell's steady state.