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Electricity and Magnetism Level 2

We are given n n resistors, each of resistance R R .The ratio of the maximum and minimum resistance that can be obtained by combining them is __________ \text{\_\_\_\_\_\_\_\_\_\_} .

log ⁡ n \log n n 2 n^2 n n n^n n n

Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

When all the resistances are connected in series,
R e q = R m a x = R + R + … R ⏟ n times = n R R_{eq} = R_{max} = \underbrace{R + R + \ldots R}_\text{n times} = nR
When all the resistances are connected in parallel,
1 R e q = 1 R m i n = 1 R + 1 R + … 1 R ⏟ n times = n R → R m i n = R n \dfrac{1}{R_{eq}} = \dfrac{1}{R_{min}} = \underbrace{\dfrac{1}{R} + \dfrac{1}{R} + \ldots \dfrac{1}{R}}_\text{n times} = \dfrac{n}{R} \rightarrow R_{min} = \dfrac{R}{n}
R m a x R m i n = n R R n = n 2 \dfrac{R_{max}}{R_{min}} = \dfrac{nR}{\dfrac{R}{n}} = n^{2}

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Moderator note:

Why is that the maximum and the minimum? E.g. can't we get a value of 0 by having a short circuit?

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