To find the potential on the Nth shell due to N concentric shells with charges \( Q, 2Q, 3Q, \ldots, NQ \) and radii \( a, 2a, 3a, \ldots, Na \), we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Problem**:
- We have N concentric shells.
- The charges on these shells are \( Q, 2Q, 3Q, \ldots, NQ \).
- The radii of these shells are \( a, 2a, 3a, \ldots, Na \).
- We need to find the electric potential at the Nth shell.
2. **Formula for Electric Potential**:
- The electric potential \( V \) due to a shell of charge \( q \) at a distance \( r \) from the center is given by:
\[
V = \frac{kq}{r}
\]
- Here, \( k = \frac{1}{4\pi \epsilon_0} \).
3. **Calculating the Potential at the Nth Shell**:
- The potential at the Nth shell due to each of the shells can be calculated individually and then summed up.
- The potential at the Nth shell due to the first shell (charge \( Q \), radius \( a \)):
\[
V_1 = \frac{kQ}{a}
\]
- The potential at the Nth shell due to the second shell (charge \( 2Q \), radius \( 2a \)):
\[
V_2 = \frac{k(2Q)}{2a} = \frac{kQ}{a}
\]
- The potential at the Nth shell due to the third shell (charge \( 3Q \), radius \( 3a \)):
\[
V_3 = \frac{k(3Q)}{3a} = \frac{kQ}{a}
\]
- Continuing this way, the potential due to the Nth shell (charge \( NQ \), radius \( Na \)):
\[
V_N = \frac{k(NQ)}{Na} = \frac{kQ}{a}
\]
4. **Summing the Potentials**:
- The total potential \( V_{nth} \) at the Nth shell is the sum of the potentials from all N shells:
\[
V_{nth} = V_1 + V_2 + V_3 + \ldots + V_N
\]
- Since each term \( V_i = \frac{kQ}{a} \), we have:
\[
V_{nth} = \frac{kQ}{a} + \frac{kQ}{a} + \frac{kQ}{a} + \ldots + \frac{kQ}{a} \quad (N \text{ terms})
\]
- This simplifies to:
\[
V_{nth} = N \cdot \frac{kQ}{a}
\]
5. **Final Expression**:
- Therefore, the potential at the Nth shell is:
\[
V_{nth} = \frac{NkQ}{a}
\]
- Substituting \( k = \frac{1}{4\pi \epsilon_0} \):
\[
V_{nth} = \frac{NQ}{4\pi \epsilon_0 a}
\]
### Conclusion:
The potential on the Nth shell due to N concentric shells having charges \( Q, 2Q, 3Q, \ldots, NQ \) and radii \( a, 2a, 3a, \ldots, Na \) respectively is:
\[
V_{nth} = \frac{NQ}{4\pi \epsilon_0 a}
\]
