Let I= current through a conductor, R= its resistance and V=...

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 $\mathrm{ML}^{2} \mathrm{I}^{-2} \mathrm{~T}^{-3}$ and $\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{I}^{-1}$ respectively.

Key Concepts

Ohm's Law

Ohm's law is a fundamental principle in electrical circuits, stating that the potential difference (voltage) across a conductor is directly proportional to the current flowing through it, with the proportionality constant being the resistance of the conductor. Through dimensional analysis, by considering the dimensions of current, voltage, and resistance, one can verify that the product of current and resistance yields the dimensional form of voltage, thus supporting the validity of Ohm's law.

Dimensional Analysis

Dimensional analysis is a method used to check the plausibility of physical equations by comparing the dimensions of the physical quantities involved. It involves expressing each quantity in terms of the basic dimensions (mass, length, time, and electric current, among others) and ensuring that both sides of an equation match dimensionally. This technique is particularly useful in verifying that a derived physical law or relationship is consistent with the underlying physical quantities.

Dimensional Consistency

Dimensional consistency refers to the requirement that all terms in a physical equation must have the same dimensions. This means that, when physical equations are written, every term must be dimensionally homogeneous. If the dimensions do not match, the equation cannot represent a physical reality. In the context of deriving laws like Ohm's law through dimensional analysis, ensuring consistency validates that the combination of quantities correctly represents the physical relationship.

Dimensional Formula

A dimensional formula expresses how a physical quantity is constructed from the base dimensions, such as mass (M), length (L), time (T), and electric current (I). It provides a compact way to describe the nature of physical quantities and is essential in dimensional analysis. For example, in the problem, the given dimensional formulas for potential difference and resistance allow us to confirm the relationship stated by Ohm's law.

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