Chapter 13: A.C Circuits - Long Answer and MCQs

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Questions and Answers

At resonance in an LC circuit, the electric current is referred to as?

The idle current

The displacement current

The wattless current (correct)

The apparent current

In a series LCR circuit at resonance, how are the applied emf and current related?

Differ in phase by π/2 radian

Differ in phase by π/4 radian

Out of phase

In phase (correct)

What is the main function of a series LCR resonant circuit?

A radio wave transmitter

A potential divider circuit

A source of displacement current

A tunning circuit in a television receiver set (correct)

When AC voltage is applied to a pure capacitor, how does the voltage across the capacitor relate to the current?

Lags the current by phase angle π/2 rad Signup and view all the answers

What is the main purpose of a parallel LC resonant circuit?

A tunning circuit in a television receiver set Signup and view all the answers

What does an electric current in an LC-circuit at resonance represent?

The wattless current Signup and view all the answers

What does Faraday's Law of electromagnetic Induction state?

The induced electromotive force (emf) is proportional to the rate of change of magnetic field strength Signup and view all the answers

What is the mathematical relation between the number of turns in the primary coil to the secondary coil in a step-up transformer?

$N_p = kN_s$ where $k$ is a constant greater than 1 Signup and view all the answers

When do we say two coils are perfectly coupled with each other?

When all magnetic field lines are linked with both coils Signup and view all the answers

What is Lenz's Law?

The induced emf always opposes the change in magnetic flux that produced it Signup and view all the answers

A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil varies from 0 to 20 A in 0.5 s, what is the change of flux linked with the other coil?

$15 , \text{Wb}$ Signup and view all the answers

An aircraft with a wing span of 50 m flies horizontally in Earth's magnetic field at a speed of 400 m/s. What is the emf generated between the tips of the wings?

$1.2 , \text{V}$ Signup and view all the answers

What is the peak value of an alternating source if an electric bulb, which operates at 10 V d.c., connected to it gives normal brightness?

28.28 V Signup and view all the answers

When a coil of resistance 300Ω and inductance 1.0 H is connected to an alternating voltage, what is the phase difference between the voltage and current in the circuit?

45° Signup and view all the answers

What is the angular frequency of free oscillations when a charged 10 micro farad capacitor is connected to an 81 mH inductor?

$1.5 \times 10^3$ rad per sec Signup and view all the answers

What is the condition for current resonance in an LCR series circuit?

Maximum current Signup and view all the answers

State one characteristic of a parallel LC AC resonance circuit.

Minimum bandwidth Signup and view all the answers

What is the relationship between average current and rms current over half a cycle in an AC circuit?

$\frac{1}{\sqrt{2}}$ Signup and view all the answers

Study Notes

LC Circuit at Resonance

At resonance, the electric current in an LC circuit is maximum and in phase with the applied voltage.
The impedance of the circuit is minimal at resonance.
Resonance occurs when the inductive reactance ($X_L$) and capacitive reactance ($X_C$) are equal, resulting in an energy exchange between the inductor and capacitor.

Series LCR Circuit at Resonance

In a series LCR circuit at resonance, the applied emf and current are in phase.
The impedance of the circuit is at its minimum, leading to a maximum current.
The circuit acts as a purely resistive circuit.

Function of a Series LCR Resonant Circuit

The main function of a series LCR resonant circuit is to selectively amplify signals at a specific frequency.
It acts as a band-pass filter allowing signals at the resonant frequency to pass through while attenuating other frequencies.

AC Voltage and Capacitor

When AC voltage is applied to a pure capacitor, the voltage across the capacitor lags behind the current by 90 degrees.
The capacitor stores energy in the form of an electric field.

Parallel LC Resonant Circuit

The main purpose of a parallel LC resonant circuit is to selectively attenuate signals at a specific frequency.
This circuit acts as a band-stop filter, blocking signals at the resonant frequency while passing other frequencies.
The impedance of the circuit is maximum at resonance.

Electrical Current in an LC Circuit

The electric current in an LC circuit at resonance represents the flow of charge between the inductor and capacitor.
This current is oscillatory and flows back and forth between the two components.

Faraday's Law of Electromagnetic Induction

Faraday's Law of electromagnetic induction states that the magnitude of the induced electromotive force (emf) in a circuit is proportional to the rate of change of magnetic flux linkage through the circuit.
The direction of the induced emf is such that it opposes the change in magnetic flux.

Step-up Transformer

The mathematical relation between the number of turns in the primary coil ($N_p$) and the secondary coil ($N_s$) in a step-up transformer is:
- $N_s > N_p$
This means the secondary coil has more turns than the primary coil, leading to increased voltage in the secondary circuit.

Perfectly Coupled Coils

Two coils are considered perfectly coupled when all magnetic flux produced by one coil links the other coil.
In this case, the mutual inductance between the coils is maximum.

Lenz's Law

Lenz's Law states that the direction of the induced current in a circuit is such that it opposes the change in magnetic flux that produces it.
It's essentially a consequence of the principle of conservation of energy.

Change in Flux Linkage

The change of flux linked with the other coil is calculated using the formula:
- $\Delta \Phi = M \Delta I$
Where M is the mutual inductance (1.5 H) and $\Delta I$ is the change in current (20 A).
The change in flux is 30 Weber.

Induced EMF in an Aircraft's Wings

The emf generated between the tips of the wings can be calculated using the formula:
- $emf = Blv$
Where B is the Earth's magnetic field strength (assumed to be constant), l is the wing span (50 m), and v is the aircraft's speed (400 m/s).
The generated emf is 20,000 V.

Peak Value of AC Source

The peak value of an alternating source is calculated by:
- $V_p = \sqrt{2} V_{rms}$
The peak value is 14.14 V.

Phase Difference in RL Circuit

The phase difference between voltage and current in an RL circuit is calculated using the phase angle:
- $tan \phi = \frac{X_L}{R}$
Where $X_L = \omega L$ is the inductive reactance, ω is the angular frequency, and R is the resistance.
The phase difference is approximately 18.43 degrees, meaning voltage leads current.

Angular Frequency of Free Oscillations

The angular frequency of free oscillations in an LC circuit is calculated by:
- $\omega = \frac{1}{\sqrt{LC}}$
Using the given capacitance (10 µF) and inductance (81 mH), the angular frequency is 1109.4 rad/s.

Current Resonance in LCR Circuit

The condition for current resonance in an LCR series circuit is:
- $X_L = X_C$
This means the inductive reactance and capacitive reactance are equal, leading to minimum impedance and maximum current.

Characteristic of Parallel LC Circuit

One characteristic of a parallel LC AC resonance circuit is it exhibits high impedance at the resonant frequency.
This is because the inductor and capacitor resonate, creating a high impedance path for the current.

Relationship between Average and RMS Current

The relationship between average current ($I_{av}$) and rms current ($I_{rms}$) over half a cycle in an AC circuit is:
- $I_{av} = \frac{2}{\pi} I_{rms}$
This means the average current is approximately 0.637 times the rms current.

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This quiz covers topics related to A.C circuits, including expressions for power in magnetic fields, Ampere's law, and transformers. It includes long answer questions as well as multiple-choice questions.