LC Oscillations Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. Which of the following is a true statement?

1. When a resistor is connected to a capacitor, the electric current in the circuit undergoes LC oscillations
2. When a resistor is connected to an inductor, the electric current in the circuit undergoes LC oscillations
3. When a charged capacitor is connected to an inductor, only the electric current in the circuit undergoes LC oscillations
4. When a charged capacitor is connected to an inductor, the electric current in the circuit and charge on the capacitor undergoes LC oscillations.

Ans. The correct option is d. When a charged capacitor is connected to an inductor, the electric current in the circuit and charge on the capacitor undergoes LC oscillations.

Explanation: When a charged capacitor is connected to an inductor, the electrical energy of the capacitor is converted into the magnetic energy of the inductor and vice-versa. This process gives rise to oscillations in the circuit, known as LC oscillations.

Ques. LED lights of chargers glow even after it is switched off. Which of the following causes this situation?

1. Phasors
2. Power of the AC circuit
3. LC Oscillations
4. Eddy currents

Ans. The correct option is c. LC Oscillations

Explanation: The energy of the capacitor and inductor keeps oscillating even after the power supply is disconnected. Therefore, LED lights of chargers glow even after it is switched off.

Ques. In LC oscillations

1. The maximum current is Q0/√(LC)
2. The time period of oscillation is 2π/√(LC)
3. The maximum potential difference across the inductor is Q₀/2C
4. The maximum rate of change of current is Q₀/3LC

Ans. The correct option is a. The maximum current is Q₀/√(LC)

Explanation: In an LC oscillation circuit, the magnetic energy stored in the inductor is equal to the electric energy stored in the capacitor. i.e. 

U_m = U_e

⇒ 1/2 LI₀² = 1/2 Q₀²/2C

Where

• I₀ is the maximum current through the inductor
• Q₀ is the maximum charge stored in the capacitor

⇒ I₀ = Q₀/√(LC)

Ques. What is an inductor?

Ans. An inductor, also known as a choke, coil, or reactor, is a passive two-terminal electrical component that typically consists of an insulated wire wound into a coil, that stores energy in a magnetic field when electric current flows through it.

Ques. What is a capacitor?

Ans. A capacitor is a passive two-terminal electrical component, that stores electrical energy in an electric field due to the accumulation of electric charges on two closely spaced insulated surfaces.

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Very Short Answers Questions [2 Marks Questions]

Ques. What is an LC circuit?

Ans. An LC circuit also known as a resonant circuit, tuned circuit, or tank circuit is a type of electric circuit that contains a capacitor of capacitance C and an inductor of inductance L.

Ques. What is an LC oscillation?

Ans. Electrical oscillations are produced by the exchange of energy between a capacitor which stores electrical energy and an inductor which stores magnetic energy are called LC oscillations.

Ques. What are the applications of LC Oscillations?

Ans. The applications of LC oscillations are as follows

• LC circuits behave as electronic resonators, which can be used in Oscillators, Filters, Amplifiers, and Tuners.
• It is used in tuning radio transmitters and receivers.
• Voltage magnification is provided by a series resonant circuit.
• Current magnification is provided by a parallel resonant circuit.

Ques. What are the reasons for the decrease in the amplitude of the oscillations LC oscillation circuit?

Ans. In an LC circuit, the resistance of the circuit plays the role of friction which decreases the amplitude of the oscillations. As energy is dissipated in the form of heat, so LC circuit becomes warmer with the passage of time. With the rise in temperature, the resistance of the LC circuit increases and hence the dissipation of the energy becomes faster. As a result of this, the amplitude of LC oscillations decreases rapidly.

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Very Short Answers Questions [3 Marks Questions]

Ques. Write the expression of the frequency of oscillation for the LC circuit.

Ans. Let a capacitor C be charged and connected to the inductor L at time t = 0.

After some time t, let the charge on the capacitor be q current I flowing in parallel LC circuit. Then

q/C = -L dI/dt

⇒ q/LC = - dI/dt

But, I = dq/dt

⇒ q/LC = - d²q/dt²

⇒ d²q/dt² + q/LC = 0

The above equation is comparable to the equation of simple harmonic motion i.e.

d²x/dt² + ω²x = 0

In comparison, we get

ω² = 1/LC

⇒ ω = 1/√(LC)

But angular frequency, ω = 2πf

Where f is the frequency of the LC oscillations.

⇒ 2πf = 1/√(LC)

⇒ f = 1/2π√(LC)

The above equation is the expression of the frequency of oscillation for the LC circuit.

Ques. An LC circuit consists of an inductor with L = 0.09 H and a capacitance of C = 4 x 10^-4 F. The initial charge on the capacitor is 5 µC, and the initial current in the inductor is zero. What is the maximum current in the inductor?

Ans. Given

• The inductance of the inductor, L = 0.09 H
• Capacitance, C = 4 x 10^-4 F
• Initial charge on the capacitor, Qi = 5 µC = 5 x 10^-6 C

For an LC oscillation circuit, we have

1/2 LI₀² = 1/2 Q_i²/C

Where I0 is the maximum current flowing through the inductor.

⇒ I₀ = Q_i/√(LC)

⇒ I₀ = (5 x 10^-6)/√(0.09 x 4 x 10^-4)

⇒ I₀ = 8.33 x 10^-4 A

Ques. An LC circuit consists of an inductor with L = 0.09 H and a capacitance of C = 4 x 10^-4 F. The initial charge on the capacitor is 5 µC, and the initial current in the inductor is zero. What is the maximum energy stored in the inductor?

Ans.Given

• The inductance of the inductor, L = 0.09 H
• Capacitance, C = 4 x 10^-4 F
• Initial charge on the capacitor, Q_i = 5 µC = 5 x 10^-6 C

For an LC oscillation circuit, we have

1/2 LI₀² = 1/2 Q_i²/C

Where 1/2 LI₀² is the maximum energy stored in the inductor.

⇒ U_m = 1/2 Q_i²/C

⇒ U_m = 1/2 (5 x 10^-6)² / (4 x 10^-4)

⇒ U_m = 3.125 x 10^-8 J

Ques. While comparing the LC oscillations with the oscillations of the spring block system, with whom the magnetic energy can be compared and why?

Ans. The magnetic energy stored in the inductor of the LC oscillation circuit is given by

U_m = 1/2 LI²

Where

• L is the inductance of the inductor
• I is the current flowing through the circuit.

Magnetic energy stored in the inductor is analogous to the kinetic energy of the spring block system i.e.

Kinetic energy = 1/2 mv²

Where

• m = mass of the block
• v = velocity of the block

They are analogous to each other because in the LC oscillation circuit, inductance L provides resistance to change in current flowing through the circuit, in the same way mass m provides resistance to change in velocity in the spring-mass system.

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Very Long Answers Questions [5 Marks Questions]

Ques. Show that the total energy is conserved during LC oscillations.

Ans. Let L be the inductance of the inductor and C be the capacitance of the capacitor in an LC oscillation circuit. If current I flow through the circuit then the magnetic energy stored in the inductor is given by

U_m = 1/2 LI²

But, I = dq/dt = d/dt (q₀ cos ωt) = - q₀ ω sin ωt

Then. U_m = 1/2 L (- q₀ ω sin ωt)² = 1/2 Lω² q₀² sin² ωt

But, ω = 1/√(LC)

⇒ ω² = 1/LC

Hence, U_m = (q₀²/2C) sin² ωt

Electrical energy stored in the capacitor in time t is given by

U_e = 1/2 CV² = 1/2 q²/C

⇒ U_e = (q₀²/2C) cos² ωt

The total energy of the LC oscillations circuit is the sum of the magnetic energy of the inductor and the electrical energy of the capacitor. Therefore

Total energy, U = U_m + U_e

⇒ U = 1/2 Lω² q₀² sin² ωt + (q₀²/2C) cos² ωt

⇒ U = (q₀²/2C) (sin² ωt + cos² ωt) = q₀²/2C

Thus the total energy in the LC circuit at any instant of time is constant. Hence total energy is conserved.

Ques. Explain the production of the electrical oscillation in the LC circuit.

Ans. Consider that the capacitor is fully charged and the energy stored in the capacitor is given by

U_e = (q₀²/2C)

Now the capacitor begins to discharge through the inductor and hence the current begins to flow through the inductor. As a result of this magnetic field is set up around it. When the capacitor is discharged completely, the energy stored in the magnetic around the inductor is given by

U_m = 1/2 LI²

In other words, the electric energy is completely converted into magnetic energy. When the magnetic energy becomes full, the capacitor begins to recharge itself in the opposite direction. Now the magnetic energy is converted into electrical energy.

This exchange of energy from electric form to magnetic form gives rise to oscillations called LC oscillations.

Ques. In an oscillating LC circuit in which C = 4.00 µF, the maximum potential across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 mA. What is the frequency of the oscillations?

Ans. Given

• The capacitance of the capacitor, C = 4.00 µF = 4 x 10^-6 F
• The maximum potential across the capacitor, V = 1.50 V
• Maximum current flowing through the inductor, I = 50 mA = 50 x 10^-3 A

In an LC oscillation circuit, the magnetic energy stored in the inductor is equal to the electric energy stored in the capacitor. i.e. 

U_m = U_e

⇒ 1/2 LI² = 1/2 CV²

⇒ L = CV²/I²

⇒ L = (4 x 10^-6 x 1.52) / (50 x 10^-3)²

⇒ L = 3.6 x 10^-3 H

The expression of the frequency of oscillation for the LC circuit

f = 1/2π√(LC)

⇒ f = 1/2π(3.6 x 10^-3 x 4 x 10^-6)

⇒ f = 1.33 kHz

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