Question

Three equal charges are placed on the three corners of a square. If the force between $q_1$ and $q_2$ is $F_{12}$ and that between $q_1$ and $q_3$ is $F_{13}$, the ratio of magnitudes $\frac{F_{12}}{F_{13}}$ is

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{2}$

Answer 1

The correct option is B. $2$.

Let the side of the square be a.

Force acting between charges $q_1$ and $q_2$ = $F_{12}=\frac{1}{4\pi {\epsilon }_0}.\frac{q^2}{a^2}$

In right angle triangle ABC, $A{C}^2=A{B}^2+B{C}^2$

$A{C}^2=a^2+a^2=2a^2$

$AC=\sqrt{2}a$

Thus the force acting between charges $q_1$ and $q_3$ = $F_{13}=\frac{1}{4\pi {\epsilon }_0}.\frac{q^2}{(\sqrt{2}a{)}^2}=\frac{1}{4\pi {\epsilon }_0}.\frac{q^2}{2a^2}$

$\frac{F_{12}}{F_{13}}=\frac{\frac{1}{4\pi {\epsilon }_0}.\frac{q^2}{a^2}}{\frac{1}{4\pi {\epsilon }_0}.\frac{q^2}{2a^2}}=\frac{2}{1}=2$.