A non conducting ring of radius `R_(1)` is charged such that the linear charge density is `lambda_(1)cos^(2)theta` where `theta` is the polar angle. If the radius is increased to `R_(2)` keeping the charge constant, the linear charge density is changed to `lambda_(2)cos^(2)theta`. The relation connecting `R_(1)`, `R_(2)lambda_(1)` and `lambda_(2)` will be

Options

A. `lambda_(1)//R_(1)=lambda_(2)//R_(2)`
B. `lambda_(1)//R_(2)=lambda_(2)//R_(1)`
C. `lambda_(1)lambda_(2)=R_(1)R_(2)`
D. `lambda_(1)^(2)R_(1)=lambda_(2)^(2)R_(2)`

Explanation

To solve the problem, we need to establish a relationship between the initial and final linear charge densities and the corresponding radii of the non-conducting ring.

### Step-by-step Solution:

1. **Understanding the Charge Distribution**:
The initial linear charge density on the ring is given by:
\[
\lambda_1 = \lambda_1 \cos^2 \theta
...