Consider an isolated cylindrical conductor A, of radius R and carrying a charge per unit length λ. We assume the conductor to be infinitely long. Consider a point P outside the conductor at a distance r from its axis. To find the electric field intensity at P, we choose a cylindrical Gaussian surface S of radius r through P and coaxial with the conductor A. As λ is the charge per unit length of conductor A, the net charge enclosed by the Gaussian cylinder of length l is
Q = λl ………….. (1)
A small element on the curved part of the Gaussian surface and containing P has an area dS.
Electric feild intensity at a point outside a uniformly cylindrical conductor assumed tp be infinitely long
Charge is uniformly distributed over the outer surface of the cylindrical conductor. Then, by symmetry, the electric field intensity at any point outside the conductor is perpendicular to the cylinder axis. Hence, the component of the electric field intensity perpendicular to the plane circular faces of the Gaussian surface is zero. Therefore, the electric flux through these flat faces is zero.
By symmetry, the electric field intensity \(\vec E\) at every point on the curved face of surface S is normal to the surface and has the same magnitude \(\vec E\). If the charge on conductor A is positive, E is directed along the outward drawn normal d.\(\vec S\).
The angle θ between \(\vec E\) and d \(\vec S\)being zero for every surface element, the electric flux
through every element is
dΦ = \(\vec E\). d\(\vec S\) = E dS
Therefore, the flux through the curved face of the Gaussian surface S is
Φ = ∮ E dS = E ∮ dS ……….. (2)
∮ dS = area of the curved surface = 2πrl, where l is the length of the cylinder as shown in the figure.
∴ Φ = E × 2πrl ………….. (3)
Then, by Gauss’s theorem,
where ε0 is the permittivity of free space and k = ε/ε0 is the relative permittivity (dielectric constant) of the surrounding medium.
This gives the magnitude of the electric field intensity in terms of the linear charge density λ. For positive λ, \(\vec E\) is outward, while for negative λ, \(\vec E\) is inward.
