Question

how will you find the direction of the net electric field at a general point due to short dipole obtain an exp for it

Answer 1

Expression for direction of electric field due to electric dipole:

The dipole has positive +q and negative -q charges separated by the distance 2d. The general point is at a distance 'r' form the midpoint 'O'.

Let the general point be 'P'.

The electric field at the point 'P' due to positive electric charge:

E₁ = 1/(4πε₀) × q/(r - d)²

The electric field at the point 'P' due to negative electric charge:

E₂ = 1/(4πε₀) × q/(r + d)²

The resultant electric field at the point 'P' is:

E = E₁ + (-E₂) = E₁ - E₂

Now,

E = (1/(4πε₀) × q/(r - d)²) - (1/(4πε₀) × q/(r + d)²)

E = q/(4πε₀) × [1/(r - d)² - 1/(r + d)²]

E = q/(4πε₀) × [4rd/(r² - d²)²]

Since, d << r, we get,

E = q/(4πε₀) × (4rd)/r⁴

E = q/(4πε₀) × 4d/r³

Now, the electric dipole moment, p = q × 2d.

∴ E = 1/(4πε₀) × 2p/r³

The dipole moment is pointing towards positive charges. Thus, the electric field and dipole moment are opposite to each other.

∴ $\vec{E}$ = - 1/(4πε₀) × $\vec{p}$/r³