Question

An infinite number of charges each equal to q are placed along the $X-$ axis at $x=1,x=2,x=4,x=8$....... and so on. The potential and field respectively due to this set of charges at the origin is :

• A. $\frac{q}{3\pi {ϵ}_0}$ and $\frac{q}{2\pi {ϵ}_0}$
• B. $\frac{q}{2\pi {ϵ}_0}$and$\frac{q}{3\pi {ϵ}_0}$
• C. $\frac{q}{4\pi {ϵ}_0}$and$\frac{q}{3\pi {ϵ}_0}$
• D. $\frac{q}{2\pi {ϵ}_0}$and$\frac{q}{4\pi {ϵ}_0}$

Answer 1

The correct option is B $\frac{q}{2\pi {ϵ}_0}$and$\frac{q}{3\pi {ϵ}_0}$

Here, the required potential $V=\frac{1}{4\pi {ϵ}_0}[\frac{q}{1}+\frac{q}{2}+\frac{q}{4}+....]$

or $V=\frac{q}{4\pi {ϵ}_0}[1+\frac{1}{2}+\frac{1}{4}+....]$

or $V=\frac{q}{4\pi {ϵ}_0}.\frac{1}{1-1/2}$ (this is GP sum, $S=\frac{a}{1-r},$ here first term $a=1$ and common ratio $r=1/2$)

$=\frac{q}{4\pi {ϵ}_0}\times 2=\frac{q}{2\pi {ϵ}_0}$

The field is $E=\frac{1}{4\pi {ϵ}_0}[\frac{q}{1^2}+\frac{q}{2^2}+\frac{q}{4^2}+....]$

or $E=\frac{q}{4\pi {ϵ}_0}[1+\frac{1}{4}+\frac{1}{16}+....]$

or $E=\frac{q}{4\pi {ϵ}_0}.\frac{1}{1-1/4}$

$S=\frac{a}{1-r},$ here first term $a=1$ and common ratio $r=1/4$)

$=\frac{q}{4\pi {ϵ}_0}\times \left(4/3\right)=\frac{q}{3\pi {ϵ}_0}$