Animated Solution for Physics - Units and Measurements: Which of the following units denotes the dimensions $[M{L}^2/Q^2]$, where $Q$ denotes the electric charge?

When faced with a dimensional analysis problem that asks you to identify a physical quantity from a given set of dimensions, the best strategy is to look at the options for clues. Here, the options are ${\text{Wb/m}}^2$, henry (H), ${\text{H/m}}^2$, and weber (Wb). All of these units belong to the realm of electromagnetism.

This tells us that we need to find a formula that connects fundamental mechanical dimensions (like mass, length, and time) with electrical dimensions (like charge or current). The target dimension is $[M{L}^2/Q^2]$. Notice how it looks suspiciously similar to the dimension of energy, $[M{L}^2{T}^{-2}]$, but with charge $Q$ instead of time $T$.

To bridge the gap between energy and electromagnetism, we can recall the formula for the magnetic potential energy stored in an inductor. The energy $U$ is given by:

where $L$ is the inductance (measured in henries) and $I$ is the electric current. This equation is perfect because it directly relates energy (which we know the dimensions of) to inductance and current.

We know that electric current $I$ is defined as the rate of flow of electric charge $Q$ over time $t$. Mathematically, this is expressed as:

Let's substitute this definition of current back into our energy equation:

Now, we are ready to translate this physical equation into a dimensional equation. The dimensions of energy $U$ are the same as work, which is $[M{L}^2{T}^{-2}]$. The dimension of charge is simply $[Q]$, and the dimension of time is $[T]$. The constant $\frac{1}{2}$ is dimensionless and can be ignored.

Equating the dimensions on both sides of our modified formula, we get:

To isolate the dimension of inductance $[L]$, we can cross-multiply. Notice what happens to the time dimensions:

The $T^{-2}$ in the numerator perfectly cancels out with the $T^2$ that we brought over from the denominator. This elegant cancellation leaves us with:

This exactly matches the dimensions given in the question! Therefore, the physical quantity that possesses these dimensions is inductance. The SI unit of inductance is the henry (H), making option (b) the correct answer.