Question

Consider the two dipoles that are located on a line with dipole moments - 5 × 10−9 𝐶. 𝑚 and 9 × 10−9 𝐶. 𝑚, Find the Electric potential at the center of the line. Now, replace the electric dipoles with the magnetic dipoles such that their dipole moments are 5 × 10−9 𝐴. 𝑚2 and 9 × 10−9 𝐴. 𝑚2 . Find the magnetic field at the center. 

Answer 1

Given,

$p_1=- 5\times10^{−9} 𝐶. 𝑚$

$p_2=9\times10^{−9} 𝐶. 𝑚$

Electric potential at the center of the line

$V=\frac{p}{4\pi \epsilon_o (r^2-a^2)}$

Hence, net potential at the center $(V)=V_1+V_2$

$=\frac{-5\times 10^{-9}}{4\pi \epsilon_o (r^2-a^2)}+\frac{9\times 10^{-9}}{4\pi \epsilon_o (r^2-a^2)}$

$=\frac{4\times 10^{-9}}{4\pi \epsilon_o (r^2-a^2)}$

Magnetic dipole at the center of dipole

$B=B_1+B_2$

$=\frac{\mu_o m_1}{4\pi (d-l)^2}+\frac{\mu_o m_2}{4\pi (d-l)^2}$

Now, substituting the values,

$=\frac{\mu_o 5\times 10^{-9}}{4\pi (d-l)^2}+\frac{\mu_o 9\times 10^{-9}}{4\pi (d-l)^2}$

$=\frac{14\times 10^{-7}\times 10^{-9}}{(d-l)^2}$

$=\frac{14\times 10^{-16}}{(d-l)^2}$