Electric Field on the Axis of a Dipole (Axial Line)

Along the dipole moment vector p, that is from the negative charge to the positive charge. On the axial line the fields of +q and -q are along the same line; the field of the nearer charge wins, so the net field points in the direction of p. This is why the axial formula has no minus sign in NCERT's magnitude form.

Yes, at the same distance r from the centre. Axial: E = 2kp / r^3. Equatorial: E = kp / r^3. So E_axial = 2 x E_equatorial. Also their directions differ: axial is parallel to p, equatorial is anti-parallel (opposite) to p. NEET loves this 2:1 ratio.

Exact axial field E = 2kpr / (r^2 - a^2)^2, where 2a is the charge separation. When r is much greater than a, (r^2 - a^2)^2 becomes r^4, giving E = 2kp / r^3. In NEET numerical questions the short-dipole form E = 2kp / r^3 is almost always intended, but read the words r >> a.

A dipole has zero net charge, so the two 1/r^2 fields nearly cancel. What survives is the small difference, which scales as 1/r^3. So a dipole field drops faster with distance than a single point charge field.

In the standard NCERT formula E = 2kp / r^3, r is measured from the centre of the dipole (midpoint of the two charges) to the point on the axis. Do not measure it from one of the charges unless you use the exact (r^2 - a^2)^2 form.

E = 2kp / r^3 = 2p / (4 pi epsilon0 r^3), where p is the dipole moment, r is the distance from the centre, and k = 1/(4 pi epsilon0). It is valid for a short dipole, r much greater than a.

It is parallel to the dipole moment p, pointing from -q to +q along the axis.

At the same distance, axial field is twice the equatorial field (2:1). Axial points along p; equatorial points opposite to p.

E = 2kpr / (r^2 - a^2)^2, with 2a the separation. For r >> a it reduces to E = 2kp / r^3.

The 2kp/r^3 axial result, its direction along p, and the 2:1 axial-to-equatorial ratio are directly tested in dipole and electrostatics questions almost every year.