Two concentric spherical shell's of radius R1 & R2 have q1 & q2 charge respectively as shown in figure. How much charge will flow through key k when it is closed :-

To determine how much charge will flow through the key K when it is closed, we need to analyze the electric field and potential differences created by the two concentric spherical shells with charges q1 and q2. Let's break this down step by step.

Understanding the Setup

We have two spherical shells: the inner shell with radius R1 and charge q1, and the outer shell with radius R2 and charge q2. Since these shells are concentric, they share the same center. When the key K is closed, we essentially create a conductive path between the two shells.

Electric Field Between the Shells

First, we need to consider the electric field in the region between the two shells (i.e., for R1 < r < R2). According to Gauss's law, the electric field E at a distance r from the center due to a charged shell is given by:

• If r < R1 (inside the inner shell), E = 0.
• If R1 < r < R2 (between the shells), E = (1/(4πε₀)) * (q1/r²).
• If r > R2 (outside both shells), E = (1/(4πε₀)) * ((q1 + q2)/r²).

Thus, in the region between the shells, the electric field is solely due to the inner shell's charge q1.

Potential Difference Calculation

The potential difference V between the two shells can be calculated by integrating the electric field from R1 to R2:

V = - ∫(E dr) from R1 to R2 = - ∫((1/(4πε₀)) * (q1/r²) dr) from R1 to R2.

Carrying out this integration gives:

V = (1/(4πε₀)) * (q1/R1 - q1/R2).

Charge Flow Through Key K

When the key K is closed, the potential difference V between the two shells will drive a charge Q through the key. The amount of charge that flows can be determined using the relationship:

Q = C * V,

where C is the capacitance of the system. For two concentric spherical shells, the capacitance C can be expressed as:

C = (4πε₀ * R1 * R2) / (R2 - R1).

Substituting this into the charge equation gives:

Q = (4πε₀ * R1 * R2 / (R2 - R1)) * (q1/R1 - q1/R2).

Final Expression for Charge Flow

After simplifying, we find that the charge Q that flows through the key K when it is closed is:

Q = (4πε₀ * q1 * R2) / (R2 - R1).

This equation shows that the amount of charge flowing through the key depends on the charge q1 on the inner shell, the radii of the shells, and the permittivity of free space ε₀.

Summary

In summary, when the key K is closed, the charge that flows through it is determined by the potential difference created by the inner shell's charge and the geometry of the two shells. The derived formula provides a clear relationship between these factors, allowing us to calculate the charge flow effectively.