Given:
- Two identical thin rings, each of radius R, are coaxially placed at a distance R.
- Charges Q1 and Q2 are uniformly spread on the two rings, respectively.

To find:
Work done in moving a charge q from the center of the ring with charge Q1 to the other ring.

Explanation:
To find the work done, we need to consider the electric potential energy of the system before and after the charge q is moved.

Step 1: Electric Potential Energy of the System
- The electric potential energy of a system of charges is given by the expression: U = k * (Q1 * q1 / r1 + Q2 * q2 / r2)
- Here, Q1 and Q2 are the charges on the two rings, q is the charge being moved, r1 is the distance between the charge q and the center of the ring with charge Q1, r2 is the distance between the charge q and the center of the other ring, and k is the electrostatic constant.

Step 2: Electric Potential Energy before Moving the Charge
- Initially, when the charge q is at the center of the ring with charge Q1, the distance r1 is equal to zero.
- Therefore, the electric potential energy of the system before moving the charge q is U1 = k * (Q1 * q / 0 + Q2 * q / r2)
- As the first term in the expression becomes undefined, we need to consider the limit as r1 approaches zero.
- Taking the limit, the electric potential energy before moving the charge q is U1 = k * (Q1 * q / R + Q2 * q / r2)

Step 3: Electric Potential Energy after Moving the Charge
- After moving the charge q to the other ring, the distance r1 becomes equal to R.
- Therefore, the electric potential energy of the system after moving the charge q is U2 = k * (Q1 * q / R + Q2 * q / (R + R))
- Simplifying the expression, U2 = k * (Q1 * q / R + Q2 * q / (2R))

Step 4: Work Done
- The work done in moving the charge q from the center of the ring with charge Q1 to the other ring is given by the difference in electric potential energy: W = U2 - U1
- Substituting the values of U1 and U2, we get W = k * (Q1 * q / R + Q2 * q / (2R)) - k * (Q1 * q / R + Q2 * q / r2)
- Simplifying further, W = k * q * (Q2 / (2R) - Q2 / r2)

Conclusion:
The work done in moving a charge q from the center of the ring with charge Q1 to the other ring is given by W = k * q * (Q2 / (2R) - Q2 / r2), where k is the electrostatic constant, Q2 is the charge on the other ring, R is the radius of the rings, and r2 is the distance between the charge q and the center of the other ring.