Question

A cube of edge a carries a point charge q at each corner.

(a) Calculate resultant force on charge 3 by charges 5. 6 and e.

(b) Show that the resultant electric force on charge 2, by the rest of charges is given

Answer 1

As per the given question,

The length of the side of the cube =a,

charges on each of the vertices = q

a)

distance between the vertices 3 and 8 =$\sqrt{a^2+a^2}=a\sqrt{2}$

distance between the vertices 3 and 6 =$\sqrt{a^2+a^2}=a\sqrt{2}$

distance between the vertices 3 and 5 = $\sqrt{a^2+(a\sqrt{2})^2}=a\sqrt{1+2}=a\sqrt{3}$

Force at the vertices 3, due to the vertices 5, 6 and 8

$F=F_{35}+F_{83}+F_{63}$

$\Rightarrow F=\dfrac{q^2}{4\pi \epsilon_o 2a^2}(\cos 45^\circ\hat{i}-\sin{45^\circ}\hat{k})+\dfrac{q^2}{4\pi \epsilon_o 2a^2}(\cos 45^\circ\hat{i}+\sin{45^\circ}\hat{j})+\dfrac{q^2}{4\pi \epsilon_o 3a^2}(\cos 45^\circ\hat{i}+\sin{45^\circ}\hat{j}+\cos 45\hat{k})$

$\Rightarrow F=\dfrac{q^2}{4\pi \epsilon_o a^2}(\dfrac{1}{\sqrt{2}}+\dfrac{1}{3\sqrt{2}})\hat{i}+\dfrac{q^2}{4\pi \epsilon_o a^2}(\dfrac{1}{\sqrt{2}}+\dfrac{1}{3\sqrt{2}})\hat{j}+\dfrac{q^2}{4\pi \epsilon_o a^2}(\dfrac{1}{\sqrt{2}}+\dfrac{1}{3\sqrt{2}})\hat{k}$

ii)

$F_{x}= \dfrac{kq^2}{a^2}+\dfrac{kq^2}{2a^2}\cos 45$

$F_{y}= \dfrac{kq^2}{a^2}+\dfrac{kq^2}{2a^2}\sin 45$

$F_{net}=\sqrt{F_x^2+F_y^2}=k_e\dfrac{q^2}{a^2}(\sqrt{2}+\dfrac{1}{2})=\dfrac{1.9q^2}{4\pi \epsilon_o a^2}$

$K_e= \dfrac{1}{4\pi \epsilon_o}$