Electric Field due to a Hollow Cylinder of Charge

Electric Field due to a Hollow Cylinder of Charge



(a) Inside the cylinder (radial distance < R) :



When we draw a Gaussian cylinder of radius r, we find that the charge enclosed by it is zero.



This is possible if E = 0

(b) Outside the cylinder (radial distance > R) :





( q_enc = λl)



Variation of E with r is also shown graphically here.

Electric field due to a Long Uniformly Charged Solid Cylinder

Field due to a Long Uniformly Charged Solid Cylinder

Let  be the volumetric charge density and R be the radius of the cylinder



Here, the field is radically away from x-axis having a cylindrical symmetry.

Case I : r > R. Consider a Gaussian cylinder of length I and radius r about x-axis. Electric field lines come out radically. Let E be the magnitude of electric field, then 

. Now by Gauss law,




Case II : r < R.





By Gauss law

Electric Field due to a Shell (hollow sphere) of charge

Electric Field due to a Shell (hollow sphere) of charge

Consider a shell having a charge Q uniformly distributed on it surface. The surface charge density is .

   where R is radius of shell.



Case 1: Field outside the shell (radial distance > R)

We enclose the shell in a Gaussian sphere.



By Gauss law



[The result is same as that of the point charge]


Case 2: Field inside the shell (radial distance < R)

As a charge enclosed by the Gaussian sphere = 0

Electric field due to an infinite linear charge distribution

Electric field due to an infinite linear charge distribution

First, we shall look for the symmetry of the electric field. The resultant of d₁ and d₂ is along y-axis.



If a long linear charge distribution is kept along x-axis, at any point, field is directed radically away from x-axis. The field has a cylindrical symmetry.

To find electric field, we enclose the distribution in a cylinder of radius r and length /



The flux linked with the cylinder is 



By Gauss law, [where λ is linear charge density]



Variation of E with r is also shown graphically here.

Electric Field due to a Uniform Sphere of Charge

Electric Field due to a Uniform Sphere of Charge

Consider a uniform spherical charge distribution in which a charge Q is uniformly distributed over the volume of a sphere of radius R.

The volumetric charge density is given by,

 (as the distribution of charge is uniform)


Case 1: Field outside the sphere (radial distance > R).
The field has spherical symmetry



By Gauss's law, 



[Note that the result is same as that of a point charge]




Case 2: Field inside the sphere (radial distance < R).

Consider a Gaussian sphere inside the sphere of charge.

Q  = total charge of sphere

Q' = charge enclosed by the Gaussian sphere

Electric field due to an infinite, thin non-conducting sheet

Electric field due to an infinite, thin non-conducting sheet

A sheet of thin plastic wrap, uniformly charged on one side can act as a thin sheet of charge.

The field has a planar symmetry. We will take a Gaussian box to enclose the charge distribution.





Applications of above result

Two parallel sheets are given surface charge densities ₁ and ₂. Electric fields in different regions are as shown



Consider the following specific cases:

(i) When ₁ = , ₂ = , the situation will be like the one shown below



(ii) When ₁ = , ₂ = - , the situation will be like the one shown below