Consider an isolated charged hollow spherical conductor A, of radius R and surface charge density σ, placed in a medium of permittivity ε. Consider a . point P outside the conductor at a distance r from its centre. To find the electric field intensity at P, we choose a spherical Gaussian surface S of radius r through P and concentric with conductor A. A small element of this surface containing P has an area dS.
The charge on the sphere is Q = σ (4πR2)....(1)
Electric feild intensity at a point outside a changed conducting sphere
The charge Q is uniformly distributed over the outer surface of the spherical conductor. Then, by symmetry, the electric field intensity at every point on surface S is normal to the surface and has the same magnitude E. If charge Q is positive, \(\vec E\) at every point on S is radially outward.
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 Gaussian surface S is
where ε0 is the permittivity of free space and k = \(\cfrac{ε}{ε_0}\) is the relative permittivity (dielectric constant) of the surrounding medium.
Equations (5) and (6) give the magnitude of the electric field intensity at a point P outside a hollow spherical conductor. If the net charge Q enclosed by the Gaussian surface is positive, \(\vec E\) is radially outward; if Q is negative, \(\vec E\) is radially inward. Equation (5) shows that for a point outside a hollow spherical conductor carrying a charge Q, the conductor behaves like a point charge Q at its centre.
Case (1) : At a point just outside the sphere, r ≅ R.
∴ E = \(\cfrac{σ}{ε}\) = \(\cfrac{σ}{kε_0}\)
Case (2) : Since electric charge resides on the outer surface of a hollow conductor, the charge inside the hollow spherical conductor is zero. Then, Einside = 0.
[Notes : (1) The surface of a charged conductor is an equipotential surface so that the electric field just outside it must be normal to the surface of the conductor. For a spherical charged conductor, it follows that the field is radial, and because the net charge is, by symmetry, uniformly distributed over its outer surface, the field is spherically symmetric, the same as for a point charge. (2) The electric field intensity is zero at all points inside a hollow charged conductor of arbitrary shape because under electrostatic condition the net charge of the charged conductor resides on its surface. This is also true for a hollow charged spherical conductor, if there is no charge in the cavity (e.g., on a conductor inside the cavity but insulated from the outer shell).]
