Bridge of resistors!

Electricity and Magnetism Level 3

There are $n + 1$ resistors in the upper and lower rows, respectively, and $n$ resistors between the rows.

The value of the total resistance between A and B can be represented as $(a \times 2^n - b)R$ , where $a$ and $b$ are positive constants.

Find $a + b$ .

The answer is 2.

1 solution

Sabhrant Sachan
Jan 4, 2017

$\dfrac{R}{2R} = \dfrac{2R}{4R} = \dfrac{4R}{8R}= \cdots = \dfrac{2^{n-1}R}{2^{n}R}= \dfrac{2^{n}R}{2^{n+1}R} = \text{ Const.}$

$\begin{aligned} \dfrac{1}{R_{AB}} & = \dfrac{1}{R+2R+4R+\cdots+2^{n}R} + \dfrac{1}{2R+4R+8R+\cdots+2^{n+1}R} \\ & = \dfrac{1}{R} \left( \frac{1}{2^{n+1}-1} + \frac{1}{2(2^{n+1}-1)} \right) \\ \dfrac{1}{R_{AB}} & = \dfrac{3}{2R(2^{n+1}-1)} \\ R_{AB} & = R \left( \frac{4}{3} \times 2^{n}-\frac{2}{3}\right) \end{aligned}$

$a+b = \dfrac{4+2}{3} = 2$

Bridge of resistors!

The answer is 2.

1 solution

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