Electric Potential Outside Conducting Cylinder

When considering a long conducting cylinder with charge density lambda (λ), the electric potential outside the cylinder can be found using Gauss's law for symmetry. This law helps us analyze the electric field and potential due to the cylindrical symmetry of the setup.

Let's imagine a Gaussian surface in the form of a cylindrical shell with radius r (greater than a) and length L. The charge enclosed by this Gaussian surface is equal to the charge on the conducting cylinder, which is lambda times the length L.

Outside the cylinder, the electric field E is directed radially away from the axis of the cylinder and can be calculated as E = λ/(2πε₀r). To determine the electric potential, we integrate the electric field from infinity to the point r, giving us the expression V(r) = -∫(λ/2πε₀r)dr = λ/(2πε₀) ln(r) + C. Here, C represents the constant of integration that accounts for the boundary conditions of the system.

Therefore, the electric potential V(r) outside a long conducting cylinder with radius a and charge density lambda (λ) can be expressed as V(r) = λ/(2πε₀) ln(r) + C.

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