Question

Two electrostatic field lines are drawn between two point charges $q_1\&q_2$ as shown in the diagram , If $\alpha \&\beta$ are planar angles. then the ratio $q_1/q_2$ will be

• A.
  $\frac{\sin \left(\alpha /2\right)}{\sin \left(\beta /2\right)}$
• B.
  $\frac{{\sin }^2\left(\alpha /2\right)}{{\sin }^2\left(\beta /2\right)}$
• C.
  $\frac{{\sin }^2\left(\beta /2\right)}{{\sin }^2\left(\alpha /2\right)}$
• D.
  $\frac{\cos \beta }{\cos \alpha }$

Answer 1

The correct option is C

$\frac{{\sin }^2\left(\beta /2\right)}{{\sin }^2\left(\alpha /2\right)}$

Let us imagine a sphere around $q_1.$ If $r\to 0$ then we can neglect the effect of $q_2$ in the vicinity of $q_1.$

$\Delta s_1=\pi r^2(1-cos\alpha ).$

No of lines passing through $\Delta s_1\Rightarrow \frac{\pi r^2}{4\pi }(1-(\cos \alpha \left)\right)\times \frac{q_1}{{ϵ}_0}$

Since lines of forces do not intersect same no of lines pass through $\Delta S_2$


$\Rightarrow \frac{\pi r^2}{4\pi }(1-\cos \alpha )\frac{q_1}{{ϵ}_0}=\frac{\pi r^2}{4\pi }(1-\cos \beta )\frac{q_2}{{ϵ}_0}\Rightarrow {\sin }^2\frac{\alpha }{2}.\times q_1={\sin }^2\frac{\beta }{2}\times q_2\Rightarrow \frac{q_1}{q_2}=\frac{{\sin }^2\beta /2}{{\sin }^2\alpha /2}$

Hence Option (c) is correct answer