Question

derive an expression for the electric field at a point P on the x-axis for which X is much larger than the distance between the charges​

Answer 1

Given that,

The electric field at a point P on the x-axis for which X is much larger than the distance between the charges​.

Let us consider two charge -q and +q separated by small distance 2d.

P is a point along the axial line of the dipole at a distance X from the midpoint O of the electric dipole.

The electric field at the point P due to +q placed at B

$E_{1}=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{q}{(X-d)^2}$...(I)

The electric field at the point P due to -q placed at A

$E_{2}=-\dfrac{1}{4\pi\epsilon_{0}}\dfrac{-q}{(X+d)^2}$...(II)

We need to calculate the magnitude of resultant of electric field

Using formula of electric field

$E=E_{1}+(-E_{2})$

Put the value into the formula

$E=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{q}{(X-d)^2}-\dfrac{1}{4\pi\epsilon_{0}}\dfrac{q}{(X+d)^2}$

$E=\dfrac{q}{4\pi\epsilon_{0}}(\dfrac{1}{(X-d)^2}-\dfrac{1}{(X+d)^2})$

$E=\dfrac{q}{4\pi\epsilon_{0}}(\dfrac{4Xd}{(X^2-d^2)^2})$

If the point P is far away from the charges, and X is much larger than the distance between the charges​

The electric field will be

$E=\dfrac{q}{4\pi\epsilon_{0}}(\dfrac{4Xd}{X^4})$

$E=\dfrac{q}{4\pi\epsilon_{0}}(\dfrac{4d}{X^3})$

Hence, The electric field will be $\dfrac{q}{4\pi\epsilon_{0}}(\dfrac{4d}{X^3})$