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Question
Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are \[{\text{ML}}^2 \text{I}^{- 2} \text{T}^{- 3}\] and \[{\text{ML}}^2 \text{T}^{- 3} \text{I}^{- 1}\] respectively.
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Solution
Dimensional formula of resistance, [R] = [ML2A−2T−3] ...(1)
Dimensional formula of potential difference, [V] = [ML2A−1T−3] ...(2)
Dimensional formula of current, I = [A]
Dividing (2) by (1), we get:
\[\frac{\left[ V \right]}{\left[ R \right]} = \frac{\left[ {ML}^2 A^{- 1} T^{- 3} \right]}{\left[ {ML}^2 A^{- 2} T^{- 3} \right]} = \left[ A \right]\]
⇒ V = IR
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