Electric Field Intensity due to Various Charge Distributions

Electric Field Intensity due to Various Charge Distributions

• It is known that the electric field intensity due to a point charge Q is given by,

• Let us consider various charge distributions.

1.   due to Line Charge

• Consider a line charge distribution having a charge density ρ_L as shown in the Fig. 2.5.1.

The charge dQ on the differential length dl is,

dQ = ρ_L dl

• Hence the differential electric field d  at point P due to dQ is given by,

• Hence the total   at a point P due to line charge can be obtained by integrating d  over the length of the charge.

• The  and dl is to be obtained depending upon the co-ordinate system used.

2.   due to Surface Charge

• Consider a surface charge distribution having a charge density ρ_S as shown in the Fig. 2.5.2.

• Hence the differential electric field d  at a point P due to dQ is given by,

• Hence the total   at a point P is to be obtained by integrating d  over the surface area on which charge is distributed. Note that this will be a double integration.

• The   and dS to be obtained according to the position of the sheet of charge and the co-ordinate system used.

3.  due to Volume Charge

• Consider a volume charge distribution having a charge density p v as shown in the Fig. 2.5.3.

• The charge dQ on the differential volume dv is,

dQ = ρ_v dv

• Hence the differential electric field d at a point P due to dQ is given by,

• Hence the total  at a point P is to be obtained by integrating d over the volume in which charge is accumulated. Note that this integration will be a triple integration.

• The  and dv must be obtained according to the co-ordinate system used.

• Thus if there are all possible types of charge distributions, then the total   at a point is the vector sum of individual electric field intensities produced by each of the charges at a point under consideration.

• where  are the field intensities due to point, line, surface and volume charge distributions respectively.

• Let us discuss and learn the method of obtaining electric field intensities under widely varying charge distributions.