question_answer

Suppose an infinite number of charges equal to q are placed along x-axis at \[\operatorname{x}=1, y= 2\,\,x=4,\,\,x=8......\] so on. The value of electric field at origin will be

A)  \[\frac{q}{2\pi {{\varepsilon }_{0}}}\]               

B)  \[\frac{q}{3\pi {{\varepsilon }_{0}}}\]

C)  \[\frac{q}{4\pi {{\varepsilon }_{0}}}\]               

D)  \[\frac{q}{8\pi {{\varepsilon }_{0}}}\]

Correct Answer: B

Solution :

As the charges are placed along the same straight line, the electric field at \[\operatorname{x}= 0\] will be directed along x-axis and its magnitude is given by \[E=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{{{1}^{2}}}+\frac{q}{{{2}^{2}}}+\frac{q}{{{4}^{2}}}+\frac{q}{{{8}^{2}}}+......\,\,\infty  \right]\] \[=\,\,\frac{q}{4\pi {{\varepsilon }_{0}}}\left[ 1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}\,\,+\,\,......\,\,\infty  \right]\] \[=\,\,\frac{q}{4\pi {{\varepsilon }_{0}}}\,\,\left( \frac{1}{1-\frac{1}{4}} \right)=\frac{q}{3\pi {{\varepsilon }_{0}}}\]