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Question: An infinite number of charges, each of charge 1 μC, are placed on the x-axis with co-ordinates x = 1...

An infinite number of charges, each of charge 1 μC, are placed on the x-axis with co-ordinates x = 1, 2, 4, 8, ....∞. 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

12000 N

Explanation

Solution

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=14πε0[1×1×10−6(1)2+1×1×10−6(2)2F = \frac { 1 } { 4 \pi \varepsilon _ { 0 } } \left[ \frac { 1 \times 1 \times 10 ^ { - 6 } } { ( 1 ) ^ { 2 } } + \frac { 1 \times 1 \times 10 ^ { - 6 } } { ( 2 ) ^ { 2 } } \right.  +1×1×10−6(4)2+1×1×10−6(8)2+....∞]\left. \ + \frac{1 \times 1 \times 10^{- 6}}{(4)^{2}} + \frac{1 \times 1 \times 10^{- 6}}{(8)^{2}} + ....\infty \right\rbrack

=10−64πε0(11+14+116+164+...∞)=9×109×10−6(11−14)= \frac{10^{- 6}}{4\pi\varepsilon_{0}}\left( \frac{1}{1} + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ...\infty \right) = 9 \times 10^{9} \times 10^{- 6}\left( \frac{1}{1 - \frac{1}{4}} \right)

=9×109×10−6×43=9×103×43= 9 \times 10^{9} \times 10^{- 6} \times \frac{4}{3} = 9 \times 10^{3} \times \frac{4}{3}= 12000 N