What is the formula for potential due to an infinite line of charge?

The potential due to an infinitely long line of charge is not absolute and is usually expressed relative to a reference point due to it diverging at infinity. We express it as a potential difference between two points.

Due to the cylindrical symmetry, it's often more useful to express the potential in cylindrical coordinates.

Here's the breakdown:

• Problem: Calculating the electric potential V at a distance s from an infinitely long line of charge with uniform linear charge density λ (charge per unit length).

• Challenge: The integral for potential diverges at infinity. This means we cannot define an absolute potential. Instead, we calculate the potential difference between two points.

• Potential Difference: Let's calculate the potential difference between a point s and a reference point s₀.

  The potential difference V(s) - V(s₀) is given by:

  V(s) - V(s₀) = - (λ / 2πε₀) ln(s / s₀)

  where:

  • V(s) is the electric potential at a distance s from the line.
  • V(s₀) is the electric potential at a reference distance s₀ from the line.
  • λ is the linear charge density (charge per unit length).
  • ε₀ is the permittivity of free space (approximately 8.854 x 10^-12 F/m).
  • ln is the natural logarithm.

• Explanation:

  • The formula shows that the potential decreases logarithmically as you move away from the line of charge.
  • The potential is defined relative to a reference point s₀. Choosing a different s₀ simply shifts the potential by a constant.
  • Due to the ln(s) dependence, the absolute potential (if we tried to define it as V(∞) = 0) would be infinite. This is why potential differences are more meaningful in this scenario.

• Important Considerations:

  • This formula is valid only for an infinitely long line of charge. In reality, all lines have a finite length. However, this is a good approximation when the distance s is much smaller than the length of the line.
  • The line of charge is assumed to be uniform, meaning that the charge density λ is constant along its length.

In summary, you can't define a single formula for the potential due to an infinite line of charge without specifying a reference point. The key is the potential difference, which is given by V(s) - V(s₀) = - (λ / 2πε₀) ln(s / s₀).