Overview:
We are given two concentric shells with different potentials, and we need to find the potential at the center of the shells. The inner shell has a potential of 10V at a radius of r, while the outer shell has a potential of 5V at a radius of 2r.

Explanation:
To find the potential at the center, we need to understand the concept of potential due to concentric shells. The potential at any point inside a shell depends only on the total charge enclosed by the shell and the distance from the center of the shell.

Step 1: Calculate the total charge enclosed by each shell:
The potential at the center of a shell is given by the formula V = kQ/r, where V is the potential, k is the electrostatic constant, Q is the total charge, and r is the distance from the center of the shell. Since the shells are concentric, the distance from the center is constant for both shells.

For the inner shell:
Given potential V1 = 10V
Charge Q1 = V1 * r (using V = kQ/r)
= 10V * r

For the outer shell:
Given potential V2 = 5V
Charge Q2 = V2 * 2r (using V = kQ/r)
= 5V * 2r
= 10V * r

Step 2: Calculate the total charge at the center:
Since the shells are concentric, the total charge at the center is the sum of the charges enclosed by each shell.

Total charge at the center = Q1 + Q2
= 10V * r + 10V * r
= 20V * r

Step 3: Calculate the potential at the center:
Now that we have the total charge at the center, we can calculate the potential using the formula V = kQ/r.

Potential at the center = Vc = kQc/r
= k(20V * r)/r
= 20V * k

Conclusion:
The potential at the center of the two concentric shells is 20V.