Animated Solution for Physics - Physics and Measurement: $L$, $C$ and $R$ represent the physical quantities inductance, capacitance and resistance, respectively. The combinations which have the dimensions of frequency are

When faced with a dimensional analysis problem involving electrical components like resistance ($R$), inductance ($L$), and capacitance ($C$), it is tempting to dive straight into the fundamental dimensions.

You might start writing out $[R]=[M{L}^2{T}^{-3}{A}^{-2}]$ and $[L]=[M{L}^2{T}^{-2}{A}^{-2}]$, and then try to multiply and divide them.

While this brute-force method works, it is a massive drain on your time and highly prone to calculation errors. In competitive exams like JEE, time is your most valuable asset.

Instead of wrestling with $M,L,T,$ and $A$, we can use a much more elegant and powerful technique: Physical Formulas. By recalling the standard equations that govern electrical circuits, we can instantly deduce the dimensions of complex combinations.

The question asks us to find the combinations that have the dimensions of frequency.

What is frequency? Dimensionally, frequency ($f$) is simply the inverse of the time period. Therefore, its dimensional formula is strictly $[T^{-1}]$.

This gives us a massive clue. If we can identify combinations of $R,L,$ and $C$ that represent time, their reciprocals will naturally represent frequency. Let's analyze the options one by one using this logic.

A Measure of Time

Let's look at the first option, $\frac{1}{RC}$.

Imagine a simple circuit containing a resistor and a capacitor. When you connect a battery, the capacitor doesn't charge instantly. It takes time.

The rate at which it charges is governed by the RC time constant, denoted by ${\tau }_{RC}=RC$.

Since $RC$ is literally a measure of time, its dimension is $[T]$. Therefore, the reciprocal, $\frac{1}{RC}$, must have the dimension of $[T^{-1}]$.

This perfectly matches the dimension of frequency! So, option (a) is absolutely correct.

Magnetic Inertia

Now, let's evaluate the second option, $\frac{R}{L}$.

Consider a circuit with an inductor and a resistor. An inductor opposes any change in current, acting like "magnetic inertia."

Because of this, the current takes time to reach its maximum steady value. This delay is characterized by the LR time constant, which is given by ${\tau }_{LR}=\frac{L}{R}$.

Since $\frac{L}{R}$ has the dimension of time $[T]$, its reciprocal, $\frac{R}{L}$, must have the dimension of $[T^{-1}]$.

Once again, this is the dimension of frequency. Thus, option (b) is also correct.

The Heartbeat of Resonance

Moving on to the third option, $\frac{1}{\sqrt{LC}}$.

What happens if we connect an inductor directly to a charged capacitor? The energy begins to oscillate back and forth between the electric field of the capacitor and the magnetic field of the inductor.

This creates an LC oscillator. The natural angular frequency of this oscillation is given by the famous resonance formula:

Since $\omega$ is an angular frequency, its dimension is inherently $[T^{-1}]$.

This means $\frac{1}{\sqrt{LC}}$ directly represents frequency. Therefore, option (c) is correct.

Finally, let's quickly glance at option (d), which is $\frac{C}{L}$.

From our previous derivations, we know that $RC$ is time and $L/R$ is time. If we multiply them, we get $LC$, which has the dimension of $[T^2]$.

Dividing $C$ by $L$ does not yield anything close to $[T^{-1}]$. It is an imposter option and can be safely eliminated.

By leveraging our knowledge of time constants and resonance, we bypassed pages of tedious algebra. The correct combinations that represent frequency are (a), (b), and (c).