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The dimensional formula of k (Coulomb's constant) is ________. Take I as the dimension of current.

Asked in GUJCET 2016 · Units and dimensions

• (1) M⁻¹L²T⁰I⁻²
• (2) M¹L³T⁴I¹
• (3) M¹L³T⁻⁴I⁻²
• (4) M¹L⁻³T⁴I²

Step-by-step solution

Why the other options are wrong

• (1) No single slip produces M⁻¹L²T⁰I⁻²; the giveaway is T⁰, which cannot be right, because [F] contributes T⁻² and [q²]=I²T² in the denominator contributes another T⁻². Sanity check: k=(Fr²)/(q²) fixes it at M¹L³T⁻⁴I⁻².
• (2) M¹L³T⁴I¹ has the mass and length exponents right but both remaining signs wrong, as if q² had been multiplied in rather than divided out -- and even then the power of the charge would be 2, not 1. Sanity check: k=(Fr²)/(q²) gives M¹L³T⁻⁴I⁻².
• (4) M¹L⁻³T⁴I² is the dimensional formula of ε₀, M⁻¹L⁻³T⁴I², with the mass exponent left positive. It comes from writing Coulomb's law upside down, k=(q²)/(Fr²), which gives 4πε₀ instead of k.

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