Question

Consider a hollow spherical conductor with total charge \(+5 e\). The outer and inner radii are \(a\) and \(b\), respectively. (a) Calculate the charge on the sphere's inner and outer surfaces if a charge of \(-3 e\) is placed at the center of the sphere. (b) What is the total net charge of the sphere?

Short answer

#Answer# The charge distribution on the inner surface of the sphere is +3e, and on the outer surface is +2e. The total net charge on the sphere remains +5e.

Step-by-step solution

Apply Gauss' Law for the inner surface of the sphere

Gauss' Law states that the enclosed electric charge within a closed surface is proportional to the total electric flux through the surface. Mathematically, it's expressed as: $$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\text{enclosed}}{\varepsilon_0}$$ When a charge of -3e is placed at the center of the sphere, the charge on the inner surface of the sphere \((\)radius \(b)\) should be equal and opposite to the charge placed at the center to maintain equilibrium. Thus, the charge on the inner surface of the shell, \(Q_1\), will be +3e.

Applying conservation of charge

The total charge on the sphere is given to be +5e. Since the charge on the inner surface is +3e, we can find the charge on the outer surface, \(Q_2\), using the conservation of charge principle: $$Q_\text{total} = Q_1 + Q_2$$ Plugging in the values: $$5e = 3e + Q_2$$

Solving for the charge on the outer surface

Now, just solve the equation for \(Q_2\): $$Q_2 = 5e - 3e = 2e$$ The charge on the outer surface of the sphere is +2e.

Calculate the total net charge on the sphere

As we have found the charge on both inner and outer surfaces of the sphere, calculating the total net charge on the sphere is straightforward. We simply sum the charge on both surfaces: $$Q_\text{net} = Q_1 + Q_2$$ Plugging in the values: $$Q_\text{net} = 3e + 2e = 5e$$ The total net charge on the sphere is +5e.

Key concepts

Electric Flux

Electric flux is a key concept in electrostatics, particularly when analyzing scenarios involving Gauss’ Law. It quantifies the amount of electric field passing through a given surface. The electric flux through a surface is expressed mathematically as \( \Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} \). This formula integrates the electric field \( \mathbf{E} \) over a surface \( d\mathbf{A} \).
Understanding electric flux helps us determine the effect of an electric field distributed over a space. For instance, when a charge is at the center of a spherical conductor, it affects the electric field uniformly outward. This scenario often simplifies calculations and illustrates how charge interacts with closed surfaces generally.

• Electric flux is proportional to the net charge within a surface.
• It forms the basis for deriving Gauss' Law.
• Helpful in simplifying problems involving symmetrical charge distributions.

Conservation of Charge

The principle of conservation of charge states that the total electric charge in an isolated system remains constant. This means charges cannot be created or destroyed; they only move from one location to another.
In the case of the hollow spherical conductor, despite adding and distributing charges on the sphere, the total charge remains constant at \(+5e\).
Here's how conservation of charge plays a critical role:

• The charge \(-3e\) at the center induces a \(+3e\) charge on the inner surface to balance.
• The remaining charge must be on the outer surface, leading to a \(+2e\) charge there.

Conservation of charge ensures the predictability of interactions within electrically neutral systems and is essential in electrostatics calculations.

Spherical Conductor

A spherical conductor is often used in problems to exploit symmetry and simplify calculations. The sphere's geometry ensures that electric fields and charge distributions are radially symmetrical.
This symmetry greatly assists in applying Gauss' Law, as it leads to uniform electric flux across its surface.
Key aspects of spherical conductors include:

• Charges within the conductor's cavity induce an equal and opposite charge on its surface.
• The induced charges help distribute the electric field evenly.

By understanding these properties, this seemingly complex system can be broken into solvable parts, illustrating the elegance and practicality of geometric symmetry in physics.

Electrostatics

Electrostatics is the study of charges at rest. It focuses on the forces, fields, and potentials related to stationary charges. Central to electrostatics is the understanding of how charges interact via electric fields.
In the exercise, we explore how charges on a conductor arrange themselves to maintain electrostatic equilibrium, where the electric field inside the conductor is zero, and any excess charge resides on the surface.
This concept is critical, as it explains:

• The behavior of charges in conductive materials.
• Why the inner surface of a hollow conductor has a charge equal and opposite to any charge placed inside it.
• How equilibrium is maintained such that there’s no net electric field inside the conductor.

Electrostatics provides vital insights into the distribution of charges and fields, necessary for understanding static charge interactions.