To determine the surface charge density on the inner and outer surfaces of the conducting spherical shell, we need to analyze the electric field and charge distribution in this system. Let's break it down step by step.
Understanding the Setup
We have two main components in this scenario:
- A solid conducting sphere with radius a and a net positive charge of 2Q.
- A conducting spherical shell with an inner radius b and an outer radius c, which has a net charge of -Q.
Electric Field Inside the Conductors
In electrostatics, the electric field inside a conductor in equilibrium is zero. This means that the electric field inside the solid sphere (radius a) and the conducting shell (between radii b and c) is zero. However, we need to consider how the charges redistribute themselves due to the presence of the other charge.
Charge Distribution on the Shell
Since the solid sphere has a charge of 2Q, it will create an electric field that influences the conducting shell. The inner surface of the shell must respond to this field. To maintain electrostatic equilibrium, the inner surface of the shell will acquire a charge that exactly cancels the electric field produced by the solid sphere within the shell. Therefore, the inner surface of the shell will have a charge of:
Q_inner = -2Q
Outer Surface Charge Calculation
The total charge on the shell is -Q. Since we have already assigned -2Q to the inner surface, we can find the charge on the outer surface by using the total charge of the shell:
Q_outer = Total Charge - Q_inner = -Q - (-2Q) = Q
Surface Charge Densities
Now that we have the charges on both surfaces, we can calculate the surface charge densities. The surface charge density (σ) is defined as the charge per unit area:
Inner Surface Charge Density
The area of the inner surface of the shell (which is a sphere) is given by:
A_inner = 4πb²
Thus, the surface charge density on the inner surface is:
σ_inner = Q_inner / A_inner = -2Q / (4πb²) = -Q / (2πb²)
Outer Surface Charge Density
The area of the outer surface of the shell is:
A_outer = 4πc²
Therefore, the surface charge density on the outer surface is:
σ_outer = Q_outer / A_outer = Q / (4πc²)
Summary of Results
In summary, the surface charge densities on the inner and outer surfaces of the conducting spherical shell are:
- Inner surface: σ_inner = -Q / (2πb²)
- Outer surface: σ_outer = Q / (4πc²)
This analysis illustrates how charges redistribute in conductors to maintain electrostatic equilibrium, ensuring that the electric field inside the conductors remains zero. Understanding these principles is crucial in electrostatics and helps in various applications, from capacitors to shielding in electronic devices.
