Illustration:

Q-1: A field of 10³ N/C acts at a point. What will be the force on a charge of 3´10^-6C and -2´10^-6C, kept there?

solution:

For +3MC

For -2MC       

Discussion:

The direction of force changes for different charges. For a positive field (due to +ve charge)
1)      The force is in direction of field on the +ve charge i.e. repulsive.
2)      The force is opposite to field direction on the –ve charge i.e. attractive.

Similar inference can be drawn for negative charges.

Electric field due to:

a)  Charged ring of charge Q and radius R.
1) The Center=0.

Derivation:
Linear charge density (l) on ring =
Consider the field at center due to any element =
But the field due to point diametrically opposite = in opposite direction.

\Net field at center = 0 (By symmetry)

2)      On the axis =

On axis of ring at distance x.

                  Derivation:

                           

                                                   Fig (5)

As obvious from the diagram the field component along the line gets added due to opposite element.

*) By substituting x=0 in 2^nd result, we can get the first result.
          *) Student should verify that the graph is of the following manner.

                                        

                                                            Fig (6)

b)      Due to a straight charged rod of length 2L with charge per unit length ‘’ at a distance ‘a’ on its perpendicular bisector.
E =
      

Derivation:
The rod is divided into infinitely small elements and the field due to symmetrically; opposite part add up as shown in figure (7).

Net field at P.

            E =

            dE =

 

               

                                      Fig (7)

     
      

Useful Tips:

1) If x>>a, E =  like a point charge.

2)  If L 

i.e. for infinite length

                                               

Question:
1)      How did the electric field cancel in one direction and add in another?
Ans:

                              

Observe the direction of electric field of two points which are symmetrically opposite.

Along the axis, net field =
Along the perpendicular,

Net field =

Similarly all fields along the perpendicular to axis cancel out.