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If n resistors, each of resistance r ohms, are connected in parallel, the equivalent resistance is R. What would be the equivalent resistance if they were connected in series?

Physics

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Answer

A\text{A}

Step-by-step solution

3 steps
  1. Determine the relationship between rr, RR, and nn for the parallel connection.

    When nn resistors, each of resistance rΩr \Omega, are connected in parallel, the equivalent resistance RR is given by the formula:

    1R=1r+1r+⋯+1r(n times)\frac{1}{R} = \frac{1}{r} + \frac{1}{r} + \dots + \frac{1}{r} \quad (\text{n times}) 1R=nr\frac{1}{R} = \frac{n}{r}

    From this equation, we can express rr in terms of RR and nn:

    r=nRr = nR
  2. Calculate the equivalent resistance if the resistors were connected in series.

    If the same nn resistors, each of resistance rΩr \Omega, were connected in series, the equivalent resistance RseriesR_{series} would be:

    Rseries=r+r+⋯+r(n times)R_{series} = r + r + \dots + r \quad (\text{n times}) Rseries=n×rR_{series} = n \times r
  3. Substitute the value of rr from Step 1 into the series resistance formula.

    Substitute r=nRr = nR into the equation for RseriesR_{series}:

    Rseries=n×(nR)R_{series} = n \times (nR) Rseries=n2RR_{series} = n^2 R

    The equivalent resistance of the combination if connected in series would be n2Rn^2R.

    Comparing this with the given options:
    [A] n2Rn^2R
    [B] R/n2R/n^2
    [C] R/nR/n
    [D] nRnR

    The calculated resistance matches option [A].

    The final answer is A\text{A}.

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