Animated Solution for Physics - Physics and Measurement: Column I gives three physical quantities. Select the appropriate units for the choices given in Column II. Some of the physical quantities may have more than one choice.

Have you ever looked at a physical quantity and wondered how many different ways you can express its unit? It's like looking at a person and realizing they are a sibling, a friend, and a student all at once! In physics, formulas are the relationships that define these identities. Let's dive into this beautiful matrix match problem and uncover the multiple identities of Capacitance, Inductance, and Magnetic Induction.

Energy and Charge

Let's start by finding the units for Capacitance ($C$). We know the energy stored in a capacitor is given by the formula:

Rearranging this to solve for $C$, we get:

Since charge $q$ is measured in Coulombs and energy $U$ in Joules, one unit for capacitance is ${\text{Coulomb}}^2{\text{-joule}}^{-1}$.

But wait, there's more! We also know the fundamental relation between charge, capacitance, and voltage:

Rearranging for $C$ gives:

This gives us another unit: ${\text{Coulomb (volt)}}^{-1}$.

Time Constants and EMF

Next, let's look at Inductance ($L$). If you recall the behavior of an $LR$ circuit, the time constant $\tau$ is defined as:

This means $L=\tau R$. Since time is in seconds and resistance in Ohms, we get the unit $\text{Ohm-second}$.

Another powerful formula involving inductance is Faraday's law for induced EMF:

Rearranging for $L$ gives:

Translating this into units, we get Volts multiplied by seconds, divided by Amperes, which is ${\text{volt-second (ampere)}}^{-1}$.

The Force on a Wire

Finally, for Magnetic Induction ($B$), we can use the magnetic force on a current-carrying wire:

Solving for $B$, we get:

Force is in Newtons, current in Amperes, and length in metres. This gives the unit ${\text{Newton (ampere metre)}}^{-1}$.

By simply recalling the fundamental formulas that connect these quantities, we've successfully matched all of them. Physics is beautifully interconnected, and dimensional analysis is the perfect tool to see those connections clearly!