Charges q, 2q, -q are given to three concentric conducting spherical shells A, B and C respectively as shown in figure. The ratio of charges on inner and outer surface of shell C will be

To determine the ratio of charges on the inner and outer surfaces of shell C, we need to analyze the behavior of electric charges in conducting materials. Conductors redistribute their charges in response to external electric fields, and this property is crucial for solving the problem at hand.

Understanding Charge Distribution

When charges are placed on conducting shells, they influence each other due to electrostatic induction. In this case, we have three concentric spherical shells: A with charge q, B with charge 2q, and C with charge -q. The key points to consider are:

• Conductors allow free movement of charges, which means that any excess charge will reside on the outer surface.
• The electric field inside a conductor in electrostatic equilibrium is zero.

Analyzing Shell C

Let’s focus on shell C, which has a total charge of -q. The charge distribution will be influenced by the charges on shells A and B. Here’s how it works:

1. **Charge on Shell A (q)**: This positive charge will induce a negative charge on the inner surface of shell B. Thus, shell B will have a charge of -q on its inner surface and a total charge of 2q, leading to a positive charge of +q on its outer surface.
2. **Charge on Shell B (2q)**: The outer surface of shell B has a charge of +q, which will induce a negative charge on the inner surface of shell C. Therefore, shell C will have a charge of -q on its inner surface.

Calculating Charges on Shell C

Now, let's summarize the charges on shell C:

• Inner surface of shell C: -q (due to induction from shell B)
• Outer surface of shell C: To find this, we need to consider the total charge of shell C, which is -q. Since the inner surface has -q, the outer surface must balance this to maintain the total charge. Therefore, the outer surface will have a charge of 0.

Finding the Ratio

Now that we have the charges on both surfaces of shell C, we can find the ratio of the charge on the inner surface to that on the outer surface:

Inner surface charge = -q

Outer surface charge = 0

The ratio of the charge on the inner surface to the charge on the outer surface of shell C is:

Ratio = (Charge on inner surface) / (Charge on outer surface) = (-q) / 0

Conclusion on the Ratio

Since division by zero is undefined, we conclude that the ratio of the charges on the inner and outer surfaces of shell C is undefined. This situation arises because the outer surface does not carry any charge, which is a unique characteristic of this configuration. Understanding these principles of electrostatics helps clarify how charges interact in conductive materials.