Question

In a LCR circuit capacitance is changed from \(\mathrm{C}\) to \(2 \mathrm{C}\). For the resonant frequency to remain unchanged, the inductance should be change from \(L\) to (a) \(4 \mathrm{~L}\) (b) \(2 \mathrm{~L}\) (c) \(L / 2\) (d) \(\mathrm{L} / 4\)

Short answer

The inductance should change from L to L/2 for the resonant frequency to remain unchanged. The correct answer is (c) L/2.

Step-by-step solution

Formulate resonance frequency formulas for the initial and final LCR circuits

For a series LCR circuit, the resonance frequency (\(f_0\)) is given by the formula: \(f_0 = \dfrac{1}{2 \pi \sqrt{L C}}\) For the initial LCR circuit, we have capacitance C and inductance L: \(f_0 = \dfrac{1}{2 \pi \sqrt{L C}}\) For the final LCR circuit, the capacitance is doubled, and we will assume the inductance changes to L': \(f_0 = \dfrac{1}{2 \pi \sqrt{L' (2C)}}\)

Equate the two resonance frequency formulas

Since the resonance frequency remains unchanged, we can equate the two formulas: \(\dfrac{1}{2 \pi \sqrt{L C}} = \dfrac{1}{2 \pi \sqrt{L' (2C)}}\)

Solve for L'

Now we need to solve for L' in terms of L. First, we can cancel out constants on both sides of the equation: \(\sqrt{L C} = \sqrt{L' (2C)}\) Now square both sides to get rid of the square roots: \(L C = L' (2C)\) Since C ≠ 0, we can divide both sides by C: \(L = 2 L'\) Finally, we solve for L': \(L' = \dfrac{L}{2}\) So the inductance should change from L to L/2 for the resonance frequency to remain unchanged. The correct answer is (c) L/2.

Key concepts

Resonance Frequency

In an LCR circuit, the resonance frequency is a crucial concept that represents the frequency at which the circuit naturally oscillates with the maximal amplitude. It occurs when the reactive components, inductance, and capacitance, balance each other out, and the overall impedance of the circuit is minimized. This is mathematically expressed by the formula:

\[f_0 = \dfrac{1}{2 \pi \sqrt{L C}}\]
where:

• \(f_0\) is the resonance frequency,
• \(L\) is the inductance, and
• \(C\) is the capacitance.

At this frequency, the circuit can efficiently transfer energy, and it is often used in applications like tuning circuits for radios and televisions.

Capacitance

Capacitance is one of the key components in an LCR circuit. It measures a capacitor's ability to store charge. The higher the capacitance, the more charge a capacitor can hold at a given voltage. It is defined as:

\[C = \dfrac{Q}{V}\]
where:

• \(C\) is the capacitance,
• \(Q\) is the electric charge, and
• \(V\) is the voltage across the capacitor.

In practice, altering the capacitance in a circuit affects its resonance frequency. For instance, doubling the capacitance in an LCR circuit will change how the circuit achieves resonance unless other adjustments, like changing the inductance, are made to counteract this change.

Inductance

In an LCR circuit, inductance refers to the property of the inductor to oppose changes in current. It is related to the magnetic field generated by the current flowing through the inductor. The greater the inductance, the larger the magnetic field and the higher the circuit's ability to resist changes in current levels. This concept is expressed as:

\[L = \dfrac{N^2 \mu A}{l}\]
where:

• \(L\) is the inductance,
• \(N\) is the number of turns in the coil,
• \(\mu\) is the permeability of the core material,
• \(A\) is the cross-sectional area of the coil, and
• \(l\) is the length of the coil.

In scenarios where the inductance needs to be adjusted, such as our exercise where capacitance was doubled, compensating by halving the inductance ensures the resonance frequency remains unchanged.

Series Circuit

A series circuit, like the one found in many LCR arrangements, is one where components are connected end-to-end, so that the same current flows through each component. This type of circuit is simple yet has unique characteristics:

• The total resistance is the sum of individual resistances.
• The sum of voltage drops across each component is equal to the total voltage applied to the circuit.
• The same current flows through all components, ensuring uniform performance in devices connected in series.

In a series LCR circuit, the inductance \(L\), capacitance \(C\), and resistance \(R\) are connected in such a manner that affects how the circuit reaches resonance. Changes in the value of one of these components—like reducing the inductance—will affect overall circuit behavior and must often be matched by adjusting other components to maintain desired characteristics such as resonance frequency.