Animated Solution for Physics - Electromagnetic Induction: An AC circuit has $R=100\Omega$, $C=2\mu \text{F}$ and $L=80\text{ mH}$ connected in series. The quality factor of the circuit is

We are given three fundamental components connected in series: a resistor with resistance $R=100\Omega$, an inductor with inductance $L=80\text{mH}$, and a capacitor with capacitance $C=2\mu \text{F}$.

The question asks us to find the Quality factor (often just called the $Q$-factor) of this circuit. The Quality factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is. In simpler terms, it tells us how "sharp" the resonance of the circuit is. A higher $Q$ means a sharper, narrower resonance peak, which is highly desirable in applications like radio tuning where you want to isolate a specific frequency.

Through derivation, this relationship simplifies beautifully to a direct formula involving only the component values:

This elegant equation is our primary tool. Notice how the resistance $R$ is in the denominator outside the root. This physically implies that increasing the resistance will heavily damp the circuit, thereby lowering the Quality factor and broadening the resonance curve.

Physics formulas demand standard SI units to yield correct results.

We convert the inductance from millihenries to Henrys: $L=80\text{mH}=80\times 10^{-3}\text{H}$

Next, we convert the capacitance from microfarads to Farads: $C=2\mu \text{F}=2\times 10^{-6}\text{F}$

Now, we substitute these pristine values into our master equation:

First, we tackle the fraction inside the square root.

Dividing the coefficients: $\frac{80}{2}=40$.

Dividing the powers of ten: $\frac{10^{-3}}{10^{-6}}=10^{-3-(-6)}=10^3$.

Combining these gives us:

The square root of $40000$ is exactly $200$. The expression now collapses into a trivial arithmetic step:

The Quality factor of the circuit is 2. This is a relatively low value, indicating that the circuit is quite damped and would have a broad response to varying frequencies.