JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Electrostatics

An infinite number of point charges, each carrying

1 μ C

charge, are placed along y-axis at

y = 1 m, 2 m, 4 m, 8 m

, ______. The total force on a 1 C point charge, placed at the origin is

x \times 1 0^{3} N

. The value of x, to the nearest integer, is ______.

[

Take

\frac{1}{4 π ε _{0}} = 9 \times 1 0^{9} Nm^{2} / C^{2}]

Answer: 12

Solution

The charges lie at

y = 2^{0}, 2^{1}, 2^{2}, \dots

metres, and each repels the charge at the origin along the same direction, so their magnitudes add directly:

F = \frac{1}{4 π ε _{0}} q_{1} q_{2} n = 0 \sum \infty \frac{1}{( 2 ^{n} ) ^{2}} = 9 \times 1 0^{9} \times (1 0^{- 6}) (1) n = 0 \sum \infty \frac{1}{4 ^{n}}

The geometric series has ratio

\frac{1}{4}

and sums to

\frac{1}{1 - \frac{1}{4}} = \frac{4}{3}

, so

F = 9 \times 1 0^{3} \times \frac{4}{3} = 12 \times 1 0^{3} N

giving

x = 12

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer