Animated Solution for Physics - Physics and Measurement: In the formula $X=5Y{Z}^2$, $X$ and $Z$ have dimensions of capacitance and magnetic field, respectively. What are the dimensions of $Y$ in SI units?

Dimensional analysis is one of the most powerful tools in a physicist's arsenal. It allows us to check the validity of equations, derive relationships between physical quantities, and, as in this problem, find the dimensions of an unknown quantity hidden within a formula.

We are given the equation $X=5Y{Z}^2$, where $X$ represents capacitance and $Z$ represents the magnetic field. Our mission is to uncover the dimensions of $Y$.

Isolating the Unknown

The first logical step is to rearrange the equation to make $Y$ the subject.

When we switch to dimensional analysis, pure numbers like $5$ disappear because they are dimensionless constants. Therefore, the dimensional equation becomes:

Now, our task is reduced to finding the dimensions of capacitance ($X$) and magnetic field ($Z$).

Decoding Capacitance ($X$)

Capacitance can sometimes be tricky to remember directly. Instead of memorizing it, let's derive it from fundamental formulas. A great starting point is the energy stored in a capacitor:

We can rearrange this for $C$:

We also know that electric potential $V$ is defined as potential energy per unit charge ($V=\frac{U}{q}$). Substituting this into our capacitance equation gives:

Now, we can easily plug in the fundamental dimensions. Charge $q$ is current times time ($[AT]$), and energy $U$ has the dimensions of work ($[M{L}^2{T}^{-2}]$).

Bringing everything to the numerator, we get:

Decoding Magnetic Field ($Z$)

Next, we need the dimensions of the magnetic field, $Z$. A classic formula involving the magnetic field is the force on a current-carrying conductor:

Rearranging for $B$:

Substituting the known dimensions for force ($[ML{T}^{-2}]$), current ($[A]$), and length ($[L]$):

Simplifying this, the $L$ cancels out:

The Final Assembly

We now have the dimensions for both $X$ and $Z$. Let's substitute them back into our master equation for $[Y]$:

First, carefully expand the square in the denominator:

Finally, use the laws of exponents to combine the terms by subtracting the denominator's powers from the numerator's powers:

- For $M$: $-1-2=-3$ - For $L$: $-2-0=-2$ - For $T$: $4-(-4)=8$ - For $A$: $2-(-2)=4$

Putting it all together, we arrive at the final dimensions for $Y$:

This perfectly matches option (c). By breaking down complex quantities into their fundamental formulas, dimensional analysis becomes a straightforward and error-free process.