Electric Field at the Equatorial Point of an Electric Dipole
A dipole is a system of two equal and opposite charges separated by a distance.
The field lines of the electric dipole can be represented as shown above.
Here we can see that the density of the electric field lines is different at different points.
The electric field where the field lines are more crowded will be stronger than the region where it is less crowded.
Here, the vertical axis (YY') at the middle of the dipole is the equatorial line.
Letâ€™s find the electric field at a point on the equator of the dipole.
We have a dipole with charges
$âˆ’q$
and
$+q$
separated by a distance
$2l$
.
Point P is at the equatorial axis of the dipole at a distance
$r$
from the centre.
Let,
$E_{1}$
and
$E_{2}$
be the electric field due to the charges
$+q$
and
$âˆ’q$
respectively.
The direction of the electric field is making an angle say
$Î¸$
with the axis of the dipole.
So, The electric field can be resolved into it's horizontal and vertical components.
The magnitude of both the charges are equal and the distance of the point P from both the charges are also equal.
Thus, the magnitude of the electric field due to both the charges will also be equal.
The vertical components of the electric field at P are equal and opposite in nature, so they will cancel each other.
Thus, the net electric field at P will be due to the horizontal component of the charges only.
The distance of the point P from both the charges can be obtained by applying the Pythagoras theorem.
So, the magnitude of the electric field can be written as,
Proceeding towards the expression for the net electric field.
The cosine of the angle from the figure of dipole can be written as,
So, the net electric field at point P can be written as shown above.
And In terms of dipole moment
$(p)$
, the net electric field can be expressed as shown above.
And this is the expression of electric field, when the point P is very far from the dipole, i.e.,
$l<<r$
.
Revision
The electric field at a point at the equator of an electric dipole is given by this equation.
For
$r>>l$
, the equation of the electric field will be this.
The End