We define a quantity called Dipole Moment for such a system such that:
where conventionally, represents the direction from –q to +q.
Axis of a dipole is the line joining –q to +q
Q. Why are we defining a dipole? Why can’t we treat it as simply two charge system? Why are we giving it special treatment?
Ans. Dipole is commonly occurring system in nature. We need to generalize our results in context with the dipole system to avoid repeated single point charge calculations using Coulomb’s Law. After these results we would be able to directly apply simplified results derived here to dipole systems.
We approach the calculation of electric field due to an electric dipole in following manner:
(a) Field of an electric dipole at points in equatorial plane:
We now find the magnitude and direction of electric field due to dipole.
and they are equal in magnitude. Note that is the unit vector along the dipole axis (from -q to + q)
Substituting the values of calculated above also, by geometry,
Now, very frequently we measure electric field at large distances from the dipole ie. r>>a
Therefore, by approximation,
We know that, by definition,
Observe, the - sign, it represents that electric field at the equator is in the opposite direction to the dipole moment of the electric dipole i.e. +q to –q.
(b) Field of an electric dipole for points on the axis:
Where is the unit vector along the dipole axis (from -q to + q)
for r >> a
As we know that, by definition of dipole moment,
(i) At point on equatorial plane ( r >> a)
(ii) At point on dipole axis (r>>a)
Note: Dipole field at large distances falls off as 1/r3.
Now, we can generalize the calculation of electric field at any general point in space due to the dipole using the above results.
Any general point in space, can be located using the polar coordinates r and 𝜃, where the origin can be placed at the center of the dipole, as shown in the above figure.
Now, for any general point P in space located at distance r from centre and inclined at an angle 𝜃 with the axis of the dipole, we can imagine components of the original dipole with dipole moment such that the P lies on the equator of one component and on the axis of the other component.
Now, Lets express our dipole moment as,
Whereis the component of the original dipole moment, such that point P is located on the axis of this dipole, i.e.
Now, at P,
We know that,
One of the component will be along the axial component of electric dipole i.e. and the other component will be along the equatorial component of electric dipole i.e.