/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Assertion In series \(L-C-R\) ci... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 21: Problem 57

Assertion In series \(L-C-R\) circuit resonance can take place. Reason Resonance takes place if inductive and capacitive reactances are equal and opposite.

Short Answer

Expert verified
The assertion is true because resonance occurs when inductive and capacitive reactances are equal and opposite, as stated in the reason.

Step by step solution

01

Understanding the Assertion

An L-C-R circuit (inductor, capacitor, resistor) can resonate when certain conditions are met. Resonance in the circuit is achieved when the overall impedance is minimized, usually implying that the circuit is able to pass the maximum current for a given voltage.
02

Understanding the Reason

Resonance occurs in an L-C-R circuit when the inductive reactance ( X_L = 蠅L ) equals the capacitive reactance ( X_C = 1/( 蠅C) ). At this point, these reactances cancel each other out because they are equal in magnitude and opposite in phase.
03

Formulating Resonance Condition

For resonance to occur, set the inductive reactance equal to the capacitive reactance: X_L = X_C. This gives the equation 蠅L = 1/( 蠅C). Solving for 蠅 (angular frequency) gives the resonant frequency: 蠅 = 1/ 鈭�( LC).
04

Conclusion

The assertion that resonance can happen in an L-C-R circuit holds true because when the inductive and capacitive reactances are equal and opposite, resonance occurs. Therefore, the reason is also correct as it provides the condition under which resonance takes place.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inductive Reactance
Inductive reactance is a key property of inductors within an LCR circuit, directly affecting the circuit's resonance behavior. It is represented by the symbol \(X_L\). This property measures how much an inductor opposes the change in current through it. The formula for inductive reactance is \(X_L = \omega L\), where \(\omega\) is the angular frequency and \(L\) is the inductance of the coil. Inductive reactance increases linearly with frequency:
  • The higher the frequency of the alternating current, the larger the inductive reactance.
  • Conversely, at lower frequencies, the reactance is smaller.
This opposition changes phase with frequency, influencing how the circuit responds to different signals.
Capacitive Reactance
Capacitive reactance, designated by \(X_C\), is the property of capacitors that affects how they impede alternating current in an LCR circuit. It defines how much a capacitor resists the flow of alternating current at a certain frequency. Mathematically, it is given by the formula \(X_C = \frac{1}{\omega C}\), where \(C\) is the capacitance and \(\omega\) is the angular frequency.
  • As the frequency increases, the capacitive reactance decreases.
  • Similarly, if the frequency decreases, the reactance increases.
This inverse relationship with frequency means that capacitors let more current pass with increasing frequency, thereby interacting with the circuit dynamics in complex ways.
Resonant Frequency
Resonant frequency is the specific frequency at which an LCR circuit naturally oscillates with maximum amplitude. It is a critical condition that ensures the circuit's efficient operation by minimizing impedance. At this frequency, the inductive reactance \((X_L)\) and the capacitive reactance \((X_C)\) are equal and opposite, effectively canceling each other out. The formula for determining this frequency is \[\omega = \frac{1}{\sqrt{LC}}\].
  • This is where the impedance of the entire circuit is at its lowest, allowing maximum current to flow.
  • At resonant frequency, the circuit behaves like a pure resistive circuit without reactive components.
Resonant frequency is a foundational concept in tuning circuits for specific frequencies in applications like radio transmission and reception.
Impedance Minimization
Impedance minimization in an LCR circuit plays a vital role when achieving resonance. Impedance is the total opposition a circuit presents to the flow of alternating current, represented by \(Z\). For an LCR circuit, the expression for impedance combines resistive (\(R\)) and reactive components (inductive and capacitive reactance) as \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
  • Resonance is achieved when \(X_L = X_C\), thus \(Z\) is minimized to simply \(R\).
  • This results in the largest possible current for a given voltage.
Impedance minimization ensures efficient power transfer and is crucial in designing devices like tuners and filters to optimize performance at specific frequencies. Understanding this concept is crucial for engineering circuits that need to handle high frequencies effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A cylindrical bar magnet is kept along the axis of a circular coil. The magnet is rotated about its axis such that north pole faces the coil. The induced current in the coil (a) is zero (b) is clockwise from magnet side (c) may be clockwise or anti-clockwise (d) is anti-clockwise from magnet side

The ratio of turns in primary and secondary coils of a transformer is \(1: 20 .\) The ratio of currents in primary and secondary coils will be (a) \(1: 20\) (b) \(20: 1\) (c) \(1: 400\) (d) \(400: 1\) A

A circular coil of diameter \(21 \mathrm{~cm}\) is held in a magnetic field of induction \(10^{-4} \mathrm{~T}\). The magnitude of magnetic flux linked with the coil when the plane of the coil makes an angle of \(30^{\circ}\) with the field is (a) \(3.1 \times 10^{-6} \mathrm{~Wb}\) (b) \(1.414 \mathrm{~Wb}\) (c) \(1.73 \times 10^{-6} \mathrm{~Wb}\) (d) \(14.14 \mathrm{~Wb}\)

A circular coil expands radially in a region of magnetic field and no electromotive force is produced in the coil. This can be because \(\quad\) [NCERT Exemplar](a) the magnetic field is constant (b) the magnetic field is in the same plane as the circular coil and it may or may not vary (c) the magnetic field has a perpendicular (to the plane of the coil component whose magnitude is decreasing suitably (d) there is a constant magnetic field in the perpendicular (to the plane of the coill direction

Two coils of self-inductances \(2 \mathrm{mH}\) and \(8 \mathrm{mH}\) are placed so close together that the effective flux in one coil is completely linked with the other. The mutual inductance between these coils is (a) \(16 \mathrm{mH}\) (b) \(10 \mathrm{mH}\) (c) \(4 \mathrm{mH}\) (d) \(6 \mathrm{mH}\)
See all solutions

Recommended explanations on Physics Textbooks

Famous Physicists

Read Explanation

Solid State Physics

Read Explanation

Translational Dynamics

Read Explanation

Oscillations

Read Explanation

Fields in Physics

Read Explanation

Physics of Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.