To reduce the resonant frequency in an LCR series circuit with a generator (a) the generator frequency should

NCERT Exemplar · Q4

Q. To reduce the resonant frequency in an LCR series circuit with a generator

(a) the generator frequency should be reduced.

(b) another capacitor should be added in parallel to the first.

(d) dielectric in the capacitor should be removed.

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Concept understanding — Resonance in AC Circuits

Resonance in AC Circuits

A series circuit containing a resistor $R$ , an inductor $L$ and a capacitor $C$ driven by an AC source exhibits resonance — a sharp condition at which the circuit responds most strongly.

The Competing Reactances

In a series RLC circuit the inductor and capacitor oppose the current in opposite senses. Their reactances are

X_L = \omega L, \qquad X_C = \frac{1}{\omega C}

where $\omega = 2\pi f$ is the angular frequency. As frequency rises, $X_L$ grows while $X_C$ shrinks. The total impedance is

Z = \sqrt{R^2 + (X_L - X_C)^2}

The Resonance Condition

At one special frequency the two reactances become exactly equal and cancel:

X_L = X_C \;\Rightarrow\; \omega_0 L = \frac{1}{\omega_0 C} \;\Rightarrow\; \omega_0 = \frac{1}{\sqrt{LC}}

The corresponding resonant frequency is

f_0 = \frac{1}{2\pi\sqrt{LC}}

At this frequency the impedance falls to its minimum, $Z = R$ (purely resistive), so the current reaches its maximum value

I_{\max} = \frac{V_{\text{rms}}}{R}

Because the reactances cancel, the source voltage and current are exactly in phase — the power factor is 1 at resonance.

Physical Picture

At resonance energy sloshes back and forth entirely between the inductor's magnetic field and the capacitor's electric field, cycle after cycle. The source only has to make up the small amount of energy lost as heat in $R$ . This is the electrical analogue of a swing pushed at its natural frequency: a small periodic drive builds a large oscillation.

Sharpness and the Q-factor

How sharply the current peaks around $f_0$ is measured by the quality factor:

Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

A large $Q$ (small $R$ ) gives a tall, narrow resonance curve — the circuit is highly selective, responding to a very narrow band of frequencies. A small $Q$ gives a broad, flat peak.

Why It Matters …

Why this formula?

Resonance in AC Circuits: Why the Key Formulas Hold

Resonance in an AC circuit occurs when the inductive reactance ( $X_L$ ) and capacitive reactance ( $X_C$ ) exactly cancel each other out. Let's build the understanding step-by-step.

1. The Core Condition for Resonance

Consider a series RLC circuit (resistor $R$ , inductor $L$ , capacitor $C$ ) driven by an AC voltage source $V = V_0 \sin(\omega t)$ .

The total impedance $Z$ of the series combination is:

Z = R + j(X_L - X_C)

where:

$X_L = \omega L$ (inductive reactance)
$X_C = \frac{1}{\omega C}$ (capacitive reactance)
$j = \sqrt{-1}$

Why resonance happens:

The circuit "wants" to let maximum current flow. The opposition to current comes from both resistance and reactance. But reactance can be negative (capacitive) or positive (inductive). When they are equal in magnitude but opposite in sign, they cancel:

X_L = X_C

This is the fundamental condition — not a formula to memorize, but a logical consequence of impedance minimization.

2. Deriving the Resonant Frequency

From $X_L = X_C$ :

\omega L = \frac{1}{\omega C}

Multiply both sides by $\omega$ :

\omega^2 L C = 1

Thus:

\omega_0 = \frac{1}{\sqrt{LC}}

Since $\omega = 2\pi f$ , the resonant frequency in hertz is:

f_0 = \frac{1}{2\pi \sqrt{LC}}

Why this makes sense:

A larger $L$ or $C$ means the circuit takes longer to "oscillate" — lower frequency.
A smaller $L$ or $C$ means faster oscillations — higher frequency.
The product $LC$ controls the natural time scale of the circuit.

3. What Happens at Resonance — Key Consequences

(a) Impedance is Minimum (Purely Resistive)

At resonance, $X_L - X_C = 0$ , so:

Z = R + j(0) = R

Why: The reactive parts cancel, leaving only the resistance. The circuit behaves like a pure resistor.

(b) Current is Maximum

From Ohm's law for AC:

I = \frac{V}{Z}

At resonance, $Z = R$ (minimum possible), so current is maximum:

I_{\text{max}} = \frac{V}{R}

Why: The opposition to current is smallest when reactance cancels.

(c) Voltage Across L and C Can Be Very Large

The voltage across the inductor:

V_L = I \cdot X_L = \frac{V}{R} \cdot \omega_0 L

The voltage across the capacitor:

V_C = I \cdot X_C = \frac{V}{R} \cdot \frac{1}{\omega_0 C}

Since $X_L = X_C$ at resonance, $V_L = V_C$ in magnitude, but they are 180° out of phase — they cancel each other in the loop.

Why this is important:

If $R$ is small, $V_L$ and $V_C$ can be many times larger than the source voltage $V$ . This is called voltage magnification — a key concept for tuned circuits and filters.

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The resonant frequency of a series LCR circuit is $f_0=\dfrac{1}{2\pi\sqrt{LC}}$ , fixed only by $L$ and $C$ (not $R$ , and not the generator's own frequency). To reduce $f_0$ , the product $LC$ must increase. Checking all four options, only (b) — adding another capacitor in parallel to the first — increases $C$ and so reduces $f_0$ ; the correct option is (b).

The resonance condition

In a series RLC circuit driven by an AC generator, the inductive and capacitive reactances are

X_L=\omega L,\qquad X_C=\frac{1}{\omega C}.

Resonance occurs at the particular angular frequency $\omega_0$ where these cancel, $X_L=X_C$ :

\omega_0=\frac{1}{\sqrt{LC}}\quad\Rightarrow\quad f_0=\frac{\omega_0}{2\pi}=\frac{1}{2\pi\sqrt{LC}}

This $f_0$ is a property of the circuit's own components $L$ and $C$ — it is completely independent of $R$ , and it is not the same thing as the generator's driving frequency (the generator frequency can be tuned to match $f_0$ , but changing the generator's frequency does not change what $f_0$ is).

Checking each option

Reduce the generator frequency. The generator's operating frequency is a separate quantity from the circuit's resonant frequency $f_0$ . Turning the generator's own frequency down changes how far the circuit is being driven from resonance, but it does not touch $L$ or $C$ , so $f_0$ itself is unchanged. Incorrect.
Add another capacitor in parallel to the first. Two capacitors in parallel combine as $C_{\text{eq}}=C_1+C_2$ , which is larger than either capacitor alone. Since $f_0\propto 1/\sqrt{LC}$ , a larger $C$ gives a smaller $f_0$ . This is exactly the reduction wanted. Correct. …

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Q.To reduce the resonant frequency in an LCR series circuit with a generator

Concept understanding — Resonance in AC Circuits

Resonance in AC Circuits

The Competing Reactances

The Resonance Condition

Physical Picture

Sharpness and the Q-factor

Why It Matters …

Why this formula?

Resonance in AC Circuits: Why the Key Formulas Hold

1. The Core Condition for Resonance

2. Deriving the Resonant Frequency

3. What Happens at Resonance — Key Consequences

(a) Impedance is Minimum (Purely Resistive)

(b) Current is Maximum

(c) Voltage Across L and C Can Be Very Large

The resonance condition

Checking each option

Method: Analysing How a Circuit Change Affects the Resonant Frequency

Steps

Q.To reduce the resonant frequency in an LCR series circuit with a generator

The resonance condition

Checking each option

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