Infinite resistors

Electricity and Magnetism Level 2

The above diagram shows a circuit with an infinite series of parallel resistors, where the $n^\text{th}$ resistor has resistance $R\times 2^{n-1}.$

If the total resistance of the circuit can be written in the form $xR$ , find $x$ .

If you believe there is no such $x,$ enter the answer $0.$

1 solution

Md Zuhair
Mar 28, 2017

Relevant wiki: Resistance

Thats clear,

Here you go, The General term for Resistance is given by $R_n= nR$ .

So you see that,

For Infinite Connection, We know that for r connections in parallel , $\dfrac{1}{R_{TOTAL}} = \displaystyle{\sum^{r}_{n=1}{\frac{1}{R_n}}}$

$\dfrac{1}{R_{\infty}} = \displaystyle{\sum^{\infty}_{n=1}{\frac{1}{R_n}}}$

$\implies \dfrac{1}{R}+\dfrac{1}{2R}+\cdots \infty$

They are in GP

So $\implies \dfrac{1}{R_{Total}} =\dfrac{2}{R}$ [INFINITE GP Formula]

$\implies R_{Total} =\dfrac{R}{2}$

So $\boxed{n=0.5}$