Question

An infinite number of charges, each of charge $1\mu C$, are placed on the x-axis with co-ordinates $x=1,2,4,8,....\infty$. If a charge of 1 C is kept at the origin, then what is the net force acting on 1 C charge

• A. 9000 N
• B. 12000 N
• C. 24000 N
• D. 36000 N

Answer 1

The correct option is B 12000 N

The schematic diagram of distribution of charges on x-axis is shown in figure below :

Total force acting on 1 C charge is given by
$F=\frac{1}{4\pi {ϵ}_0}[\frac{1\times 1\times {10}^{-6}}{(1{)}^2}+\frac{1\times 1\times {10}^{[}-6]}{\left(2^2\right)}+\frac{1\times 1\times {10}^{-6}}{(4{)}^2}+\frac{1\times 1\times {10}^{-6}}{(8{)}^2}+....\infty ]$
$=\frac{{10}^6}{4\pi {ϵ}_0}(\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...\infty )=9\times {10}^9\times {10}^{-6}\left(\frac{1}{1-\frac{1}{4}}\right)=9\times {10}^9\times {10}^{-6}\times \frac{4}{3}=9\times {10}^3\times \frac{4}{3}=12000N$