13.3 Electric field at any point

  Electrostatics

To get the electric field at a general point due to a dipole, we use the earlier results.

We find electric field at A in terms of r and $\theta $ $\left( {r \gg a} \right)$.

The dipole moment can be vectorially resolved as:

The electric field at A due to $\left( {\overrightarrow p \;\cos \;\theta } \right)$ component $ = \frac{{2k\left( {p\;\cos \;\theta } \right)}}{{{r^3}}}$

Electric field at A due to $ = \frac{{k\left( {p\;\sin \;\theta } \right)}}{{{r^3}}}$

$$ \Rightarrow \quad \quad {E_R} = \sqrt {{E^2}_{p\;\cos \;\theta } + {E^2}_{p\;\sin \;\theta }} = \frac{{kp}}{{{r^3}}}\sqrt {4{{\cos }^2}\theta + {{\sin }^2}\theta } $$

Resultant, $${E_R} = \frac{{kp}}{{{r^3}}}\sqrt {3\;{{\cos }^2}\theta + 1} \quad ;\;\tan \alpha = \frac{{{E_{p\;\sin \;\theta }}}}{{{E_{p\;\cos \;\theta }}}} = \frac{{\tan }}{2}$$