Question

Two charges each +4 μC are on the x-axis, one at the origin and the other at x = 8 m.

Find the electric field on the x-axis at:

A) x = -2 m

B) x = 2 m

C) x = 6 m

D) At what point on the x-axis is the electric field zero?

E) Sketch E_x versus x

Answer 1

Here given that the 2 charges of equal magnitudes placed at point A() and at point B().

Magnitude of the charges,

A)

At point,C()(As shown in the figure)

Here,the electric field at C by the charge at A,(),

The distance between the point C and A,

So,(directed towards -x axis,as the electric field is directed away from the +ve charge)

Also,the electric field at C by the charge at B,(),

The distance between the point C and B,

So,(Here also,electric field directed towards -x axis,as the electric field is directed away from the +ve charge)

So,the net electric field at point C(),

(directed towards -x axis as shown in the figure)

B)

Here the point C()

Here,the electric field at C by the charge at A,(),

The distance between the point C and A,

So,

(directed towards +x axis,as the electric field is directed away from the +ve charge)

Also,the electric field at C by the charge at B,(),

The distance between the point C and B,

So,(Here also,electric field directed towards -x axis,as the electric field is directed away from the +ve charge)

So,

So,the net electric field at point C(),

(directed towards +x axis as shown in the figure)

C)

Here the point C()

Here,the electric field at C by the charge at A,(),

The distance between the point C and A,

So,

(directed towards +x axis,as the electric field is directed away from the +ve charge)

Also,the electric field at C by the charge at B,(),

The distance between the point C and B,

So,(Here also,electric field directed towards -x axis,as the electric field is directed away from the +ve charge)

So,

So,the net electric field at point C(),

(directed towards -x axis as shown in the figure)

D)

Now let us consider the point C ,at a distance from the origin,where the net electric field is zero,ie,

Here,the electric field at C by the charge at A,(),

The distance between the point C and A,

So,

(directed towards +x axis,as the electric field is directed away from the +ve charge)

Also,the electric field at C by the charge at B,(),

The distance between the point C and B,

So,(Here also,electric field directed towards -x axis,as the electric field is directed away from the +ve charge)

So,

So,the net electric field at C,

So,

implies, or,

So,

So,the net electric field will be zero at the point C() ,(At the mid point of the 2 charges)

E)

Let consider the the charge placed at the origin.

We have the electric field at a distance fro the charge as,

So, (Since the others remains constant)

So,the graph of versus is shown below,