GITAM, Department of Engineering Physics

 

Applications of Gauss' Law

Gauss' law is a powerful tool for the calculation of electric fields when they originate from charge distributions of sufficient symmetry to apply it.

 

If the charge distribution lacks sufficient symmetry for the application of Gauss' law, then the field must be found by summing the point charge fields of individual charge elements. Examples are:

 

Electric Field of Line Charge

Let us consider a uniformly – charged (say positive) wire of infinite length having a constant linear charged density (i.e. charge per unit length) l.  Let ‘p’ be a point distant ‘z’ from the wire at which the electric intensity E is required. The electric field of a line of charge can be found by superposing the point charge fields of infinitesmal charge elements.

The integral required to obtain the field expression is

 

Electric Field: Ring of Charge

The electric field of a ring of charge on the axis of the ring can be found by superposing the point charge fields of infinitesmal charge elements. The ring field can then be used as an element to calculate the electric field of a charged disc.

The electric fields in the xy plane cancel by symmetry, and the z-components from charge elements can be simply added.

If the charge is characterized by an area density and the ring by an incremental width dR' , then:

This is a suitable element for the calculation of the electric field of a charged disc.

 

Electric Field:Disc of Charge

The electric field of a disc of charge can be found by superposing the point charge fields of infinitesmal charge elements. This can be facilitated by summing the fields of charged rings. The integral over the charged disc takes the form