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  4. An infinite number of point charges, each carrying 1 μ C charge, are placed along the y-axis at y =1 m , 2 m , 4 m , 8 m . ldots ldots The total force on a 1 C point charge, placed at the origin, is x × 103 N. The value of x, to the nearest integer, is. [Take .(1/4 π ϵ0)=9 × 109 Nm 2 / C 2]

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Q. An infinite number of point charges, each carrying $1 \mu C$ charge, are placed along the y-axis at $y =1\,m , 2\,m , 4\,m , 8\,m . \ldots \ldots$ The total force on a 1 C point charge, placed at the origin, is $x \times 10^{3} N$. The value of $x$, to the nearest integer, is_______.
[Take $\left.\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} Nm ^{2} / C ^{2}\right]$

JEE MainJEE Main 2021Electrostatic Potential and Capacitance

Solution:

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$F = k (1 C )(1 \mu C )\left[1+\frac{1}{2^{2}}+\frac{1}{4^{2}}+\frac{1}{8^{2}}+\ldots\right]$
$=9 \times 10^{3}\left[\frac{1}{1-\frac{1}{4}}\right]=12 \times 10^{3} N$