Question

Three concentric spherical shells have radii a, \(b\) and \(c(a

Short answer

(D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

Step-by-step solution

Write the expression for potential

For a spherical shell with a radius r and surface charge density σ, the potential V can be written as: \[V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}\] where Q is the total charge on the shell, and \(\epsilon_0\) is the vacuum permittivity. For our problem, we will denote the total charge for each shell as \(Q_A\), \(Q_B\), and \(Q_C\).

Find the total charge on each shell

The total charge on a spherical shell is given by: \[Q = \sigma \times \text{Surface Area}\] For each shell, their total charges are: - \(Q_A = \sigma \times 4\pi a^2\) - \(Q_B = -\sigma \times 4\pi b^2\) - \(Q_C = \sigma \times 4\pi (a+b)^2\)

Calculate the potentials for each shell

Now, we can substitute the corresponding total charges into the expression for potential to calculate the potentials for each shell: - \(V_A = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi a^2}{a} = \frac{\sigma a}{\epsilon_0}\) - \(V_B = \frac{1}{4 \pi \epsilon_0} \frac{-\sigma \times 4\pi b^2}{b} = -\frac{\sigma b}{\epsilon_0}\) - \(V_C = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi (a+b)^2}{(a + b)} = \frac{\sigma (a+b)}{\epsilon_0}\)

Compare the potentials

Now, we can compare the potentials of the shells: - \(V_C = \frac{\sigma (a+b)}{\epsilon_0} = \frac{\sigma a}{\epsilon_0} + \frac{\sigma b}{\epsilon_0} = V_A - V_B\) From this result, we can see that \(V_C = V_A \neq V_B\), which corresponds to option (D). Therefore, the correct answer is: (D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

Key concepts

Spherical Shells

In electrostatics, the concept of spherical shells is vital to understanding electric fields and potentials. A spherical shell is essentially a hollow ball. Think of it as a thin surface that is perfectly round, just like the Earth's outer layer or the shell of a ball.
When dealing with spherical shells, a crucial aspect is how they distribute charges evenly across their surfaces. Because of their symmetry, these shells allow for straightforward calculations in electrostatics. The electric field outside a charged spherical shell acts as if all the charge were concentrated at the center, simplifying our understanding of electric potentials and forces.

• In this context, using mathematical models, we can predict how charges behave on spherical shells, impacting surrounding electric fields and forces.
• In practical terms, this applies to various technologies like capacitors and shielding in electronics.

Surface Charge Density

Surface charge density, symbolized as \(\sigma\), refers to the quantity of charge per unit area on a surface. It's a key variable when calculating electric fields and potentials.
For a spherical shell, the surface charge density is uniform due to the symmetry of the shape, meaning it has the same value at every point on the shell's surface. The spherical shape facilitates even distribution, crucial for simplifying calculations in electrostatics.

• This uniform distribution helps in calculating the total charge \(Q\) using the formula \(Q = \sigma \times \text{Surface Area}\).
• The surface area of a sphere is given by \(4\pi r^2\), where \(r\) is the radius. Thus, multiplying this by \(\sigma\) provides the overall charge contained on the shell.

Electric Potential

Electric potential, often referred to as voltage, is the electric potential energy per charge at a point in a field. It is a scalar quantity and can be thought of as the electric "pressure" at a particular location.
The potential for a spherical shell can be determined using the formula \[V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}\] where:

• \(V\) is the potential,
• \(Q\) is the total charge on the shell,
• \(r\) is the radius of the shell,
• \(\epsilon_0\) is the permittivity of free space.

This relationship helps us understand how the potential varies depending on the radius of the shell and the charges present.
For multiple shells, like the concentric ones in the original problem, the potentials can be additive depending on configuration, leading us to differences and similarities, as calculated for \(V_A\), \(V_B\), and \(V_C\).

Concentric Shells

Concentric shells are spherical shells that share the same center but have different radii. They are like nested balls, each fitting perfectly inside the next.
This arrangement is fundamental in many electrostatics problems involving multiple charges and potentials. The exercise discussing concentric shells with different surface charge densities highlights how they interact with each other in terms of electric potential.

• With concentric shells, charge distribution on each shell impacts neighboring shells and their resulting potentials.
• In practical applications, these systems can be used to create capacitors, with layers of conductive materials separated by insulators.

Calculating the potentials of concentric shells involves analyzing how each shell contributes to the total electric potential field, which provides insights into their physical and engineering applications.