Answer to Question #151449 in Electricity and Magnetism for Vijay Kumar

Question

Consider three charges, one of magnitude +2Q and other two have magnitude -Q. They are
placed in a constellation as depicted in the figure below. Find out the electric intensity at some
spherical point P(r, theta,

Answer 1

Electric potential at the point P due to the charge -Q on the z axis $V_1=\frac{-Q}{4\pi \epsilon_o (r-\frac{d\cos\theta}{2})}$

Electric potential at the point P due to the charge -Q on the -z axis $V_2=\frac{-Q}{4\pi \epsilon_o (r+\frac{d\cos\theta}{2})}$

Electric potential at the point P due to the charge 2Q, $V_3=\frac{2Q}{4\pi \epsilon_o r}$

Hence, net electric potential at the point P,

$V=V_1+V_2+V_3 =\frac{-Q}{4\pi \epsilon_o (r-\frac{d\cos\theta}{2})}+\frac{-Q}{4\pi \epsilon_o (r+\frac{d\cos\theta}{2})}+\frac{2Q}{4\pi \epsilon_o r}$

$=\frac{2Q}{4\pi \epsilon_o r}-\frac{2Qr}{4\pi \epsilon_o (r^2-\frac{d^2\cos\theta}{4})}$

Hence, electric field intensity at the point P,

$E=\hat{r}\frac{\partial}{\partial r}[\frac{-Q}{4\pi \epsilon_o (r^2-\frac{d^2\cos\theta}{4})}]-\hat{\theta}\frac{1}{r}\frac{\partial}{\partial \theta}[\frac{-Q}{4\pi \epsilon_o (r^2-\frac{d^2\cos\theta}{4})}]-\hat{r}\dfrac{\partial}{\partial r}[\frac{2Q}{4\pi \epsilon r^2}]$

$E=\hat{r}[\frac{-Q(r^2+\frac{d^2\cos\theta}{4})}{4\pi \epsilon_o (r^2-\frac{d^2\cos\theta}{4})} +\frac{2Q}{4\pi \epsilon r^2}]-[\frac{-2Qd^2\times 2\sin\theta.\cos\theta}{4\pi \epsilon_o \times 4(r^2-\frac{d^2\cos\theta}{4})}]\hat{ \theta}$