# Calculate Electric Potential Between Two Cylinders

## How can we calculate the electric potential between two cylinders?

Given a long metal cylinder with radius a and supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is λ, and there is an equal negative charge per unit length on the outer cylinder.

## What is the formula for the potential V(r) for r less than b?

What is the potential difference between the two cylinders if the outer cylinder has no net charge?

## Calculating Electric Potential Between Two Cylinders

To calculate the electric potential between two cylinders, we can use the formula for the potential V(r) for r less than b, which is V(r) = (λ/2πϵ0)ln(b/a) - (λ/2πϵ0)ln(r/a). This formula takes into account the charge per unit length λ, the radii a and b, and the permittivity of free space ϵ0.

## Potential Difference and Formula

The potential of the inner cylinder with respect to the outer is Vab = (λ/2πϵ0)ln(b/a). If the outer cylinder has no net charge, the potential difference between the two cylinders is also Vab, given by (λ/2πϵ0)ln(b/a).

Calculating the electric potential between two cylinders involves understanding the distribution of charges and the distances involved. In this scenario, we have an inner cylinder with a positive charge per unit length and an outer cylinder with an equal negative charge per unit length.

The formula for the potential V(r) for r less than b takes into account the logarithmic relationship between the radii a and b. By substituting the relevant values into the formula, we can calculate the electric potential at different distances from the axis of the cylinders.

When considering the potential difference between the two cylinders, it is important to note the symmetry of the system. If the outer cylinder has no net charge, the potential difference simplifies to the potential of the inner cylinder with respect to the outer, as the charges on the outer cylinder do not contribute to the potential difference.