question_answer

Two point charges \[-q\] and \[+q/2\] are situated at the origin and the point \[(a,\,\,0,\,\,0)\] respectively. The point along the\[x-axis\], where the electric field vanishes is

A) \[x=\sqrt{2}a\]                 

B) \[x=\frac{a}{\sqrt{2}}\]

C)  \[x=\frac{\sqrt{2}a}{\sqrt{2}-1}\]                            

D)  \[x=\frac{\sqrt{2a}}{\sqrt{2}+1}\]

Correct Answer: C

Solution :

The given situation can be shown as Suppose the field vanishes at a distance\[x\], we have                 \[\frac{kq}{{{x}^{2}}}=\frac{kq/2}{{{(x-a)}^{2}}}\] \[\Rightarrow \]               \[2{{(x-a)}^{2}}={{x}^{2}}\] or            \[\sqrt{2}(x-a)=x\] \[\Rightarrow \]               \[\sqrt{2}x-\sqrt{2}a=x\] \[\Rightarrow \]               \[\sqrt{2}x-x=\sqrt{2}a\] \[\Rightarrow \]               \[(\sqrt{2}-1)x=\sqrt{2}a\] \[\Rightarrow \]               \[x=\left( \frac{\sqrt{2}a}{\sqrt{2}-1} \right)\]