Which one of the following expressions has dimensions of energy (here V is the voltage across a resistor of resistance R and I is the current through the resistor, and t is the time)?
V<sup>2</sup> / I t
V<sup>2</sup> / R t
I<sup>2</sup> / R t
I<sup>2</sup> / V t
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC Geoscientist – 2023
Option A: V²/I t – The dimensions of V/I are R (resistance), so V²/I t is dimensionally (V/I) * (V/t) * t = R * (V/t) * t. This doesn’t directly yield energy dimensions.
Option B: V²/R t – V²/R represents power dissipated in a resistor. Multiplying power by time (t) gives energy. Thus, (V²/R) × t has the dimensions of energy.
Option C: I²/R t – I²R represents power dissipated in a resistor. I²/R is dimensionally Current² / Resistance, which is not power. I²R * t would be energy.
Option D: I²/V t – I²/V is dimensionally Current² / Voltage. This does not represent power.
– Power in a resistor = V²/R = I²R = VI.
– Dimensions of Energy = [Force × Distance] = [MLT⁻² × L] = [ML²T⁻²].
V has dimensions [ML²I⁻¹T⁻³]. R has dimensions [ML²I⁻²T⁻³]. I has dimensions [I]. t has dimensions [T].
Option B: (V²/R) * t = ([ML²I⁻¹T⁻³]² / [ML²I⁻²T⁻³]) * [T] = ([M²L⁴I⁻²T⁻⁶] / [ML²I⁻²T⁻³]) * [T] = [ML²T⁻³] * [T] = [ML²T⁻²], which are the dimensions of energy.
Option C: (I²/R) * t would have been energy. I²/R * t as written in option C is [I²] / ([ML²I⁻²T⁻³] * [T]) = [I²] / [ML²I⁻²T⁻²] = [M⁻¹L⁻²I⁴T²].