L.C Circuit Fundamentals

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Questions and Answers

What happens to the capacitor when the switch in the LC circuit is turned off?

The capacitor maintains a constant voltage.

The capacitor gets discharged, resulting in a decrease in charge. (correct)

The capacitor creates a back EMF.

The capacitor continues to charge indefinitely.

What does the equation ε = -L represent in an LC circuit?

Potential difference across the capacitor.

Charge remaining in the capacitor.

Energy stored in the capacitor.

Back electromotive force generated due to a change in current. (correct)

In the context of damped oscillations, what is the damping force dependent on?

The distance from the equilibrium position.

Energy stored in the capacitor.

The velocity of the oscillatory body. (correct)

The mass of the oscillatory body.

How is the potential difference V across the capacitor determined?

V is proportional to the charge and inversely proportional to the capacitance. Signup and view all the answers

What characterizes a damped oscillator?

Its amplitude decreases due to resistive forces. Signup and view all the answers

In the general equation of SHM for an LC circuit, what are the components involved?

The periodic execution of energy between electric and magnetic fields. Signup and view all the answers

What role does the damping constant 'b' play in the analysis of damped oscillations?

It characterizes the damping force as a ratio of force to velocity. Signup and view all the answers

What does the equation Fnet = -kx - bv represent in a damped oscillator?

Net force resulting from both spring force and damping force. Signup and view all the answers

What is the relationship between the frequency of the oscillator and the driving force?

The oscillator oscillates with the same frequency as that of the periodic force. Signup and view all the answers

What happens to the amplitude of a driven oscillator as the difference between $ω_0$ and $ω$ becomes very small?

The amplitude of forced oscillation increases. Signup and view all the answers

What does the phase difference 'δ' between the oscillator and the driving force indicate?

The oscillator lags behind the driving force. Signup and view all the answers

How does the damping coefficient 'β' affect the duration of transient beats in the oscillator?

Higher damping results in shorter transient beats. Signup and view all the answers

When the oscillator frequency $ω$ is equal to its natural frequency $ω_0$, what is the phase difference 'δ'?

δ = 0 Signup and view all the answers

In the equations presented, how can 'f0cos(ωt)' be expressed in terms of the phase shift?

As a cosine function with a phase shift of –δ. Signup and view all the answers

When examining the conditions for high driving force, what can be inferred about the relationship between amplitude and mass?

Amplitude is inversely proportional to the mass. Signup and view all the answers

What is the primary characteristic of polarised light?

Its vibrations are confined to a single line. Signup and view all the answers

Which type of polarised light has its vibrations confined to a single linear direction?

Linearly polarised light Signup and view all the answers

What happens to the resultant light vector in circularly polarised light?

It rotates with constant magnitude. Signup and view all the answers

How does the intensity of the light compare in ordinary light across the crystal plate?

It varies across different positions. Signup and view all the answers

What distinguishes elliptically polarised light from circularly polarised light?

It has varying magnitude during rotation. Signup and view all the answers

What does the star symbol represent in the context of light?

Unpolarised light. Signup and view all the answers

Which of the following statements about ordinary light is true?

The vibrations are not confined to a specific direction. Signup and view all the answers

In what condition can two plane polarised waves lead to circularly polarised light?

When they superpose with phase difference. Signup and view all the answers

What phenomenon occurs when the dimension of an obstacle is comparable with the wavelength of incident light?

Diffraction Signup and view all the answers

Which type of diffraction involves no lenses being used to make the rays converge or parallel?

Fresnel Diffraction Signup and view all the answers

In Fraunhoffer Diffraction, what shape is the incident wave front typically?

Plane Signup and view all the answers

What is the primary characteristic of Fresnel Diffraction in relation to the distance from the obstacle?

The distance is finite Signup and view all the answers

What is the path difference for the deviated ray 'BK' during Fraunhoffer diffraction?

$e ext{sin} heta$ Signup and view all the answers

In Fresnel Diffraction, what happens to the light intensity on either side of the shadow?

It diminishes Signup and view all the answers

What is the relationship between destructive interference and the bright central maxima in Fresnel Diffraction?

Destructive interference causes the brightest central maxima Signup and view all the answers

What type of light source is typically used in Fraunhoffer diffraction experiments?

Monochromatic light Signup and view all the answers

What is the path difference required for a plate to act as a quarter wave plate?

$rac{ ext{λ}}{4}$ Signup and view all the answers

What is the main purpose of the first prism in the setup described?

To polarize the light Signup and view all the answers

Under what condition does the nicol prism function effectively?

When the incident beam is slightly convergent or divergent Signup and view all the answers

Which type of crystal allows the O-ray to travel at a lesser velocity than the E-ray?

Positive crystal Signup and view all the answers

What must the angle of incidence be confined to for the nicol prism to work correctly?

Confined within 14° Signup and view all the answers

What happens to the intensity of light when the second nicol is rotated back to parallel with the first?

Intensity becomes maximum Signup and view all the answers

What does a quarter wave plate primarily produce?

Circularly and elliptically polarized light Signup and view all the answers

If $ u_O > u_E$, what kind of crystal is being referred to?

Negative crystal Signup and view all the answers

Study Notes

L.C Circuit

An L.C circuit is a combination of an inductor "L" and a capacitor "C" connected to a DC source.
When the key is pressed, the capacitor charges and stores a charge of "+Q" and "-Q" with a potential difference between the plates of "V=q/c".
As the switch is turned off, the capacitor discharges, causing a decrease in "q" (charge).
The current at this point is given by "I=dq/dt".
As "q" decreases, the electric field energy stored in the capacitor decreases, transferring to the magnetic field around the inductor.
When the capacitor charge reaches zero, the energy in the capacitor is also zero.
Even though the charge is zero, the current is still zero at that specific time.
The potential difference across the capacitor plates at any instance is given by "V = q/c".
As the current flowing through the inductor increases, the magnetic field strength also increases, resulting in magnetic lines of force cutting or linking with the inductor.
This creates a back electromotive force (ε) given by "ε = -L(dI/dt)".
Applying Kirchhoff's Voltage Law (KVL) to the L.C circuit, v - ε = 0, which simplifies to "v + L(dI/dt) = 0", representing the general equation of Simple Harmonic Motion (SHM).
In this circuit, energy periodically oscillates between the electric field of the capacitor and the magnetic field of the inductor.
The L.C oscillation acts as a source of electromagnetic waves.
The angular frequency of the oscillation is given by ω = √(1/LC).
The period of oscillation is T = 2π√(LC).

Damped Oscillation

In a free oscillation, energy remains constant, causing indefinite oscillation.
In reality, due to damping forces like friction and resistance, the amplitude of an oscillatory system gradually decreases.
Oscillators experiencing this decrease are called damped oscillators, and their oscillations are known as damped oscillations.
The damping force always acts opposite to the direction of the oscillatory body's motion and is velocity-dependent.
The damping force is given by Fdam = -bv, where "b" is the damping constant (a positive value representing damping force per velocity).
The net force acting on the oscillator is Fnet = Fresistive + Fdam.
Considering a spring-mass system with damping, the net force is Fnet = -kx - bv.
Using Newton's Second Law, we can write M(d²x/dt²) = -kx - bv, which represents the equation of motion for a damped harmonic oscillator.
The equation can be rewritten as "M(d²x/dt²) + b(dx/dt) + kx = 0".
Introducing the damping coefficient β = b/(2M), the equation becomes (d²x/dt²) + 2β(dx/dt) + ω0²x = 0, where ω0 is the natural frequency of the undamped oscillator.
The solution for the displacement "x(t)" is a combination of the complementary function (xc(t)) and the particular integral (xp(t)).
The complementary function reflects the natural oscillations of the system and is given by xc(t) = Ae^(-βt)cos(ωt + φ), where A and φ are constants determined by initial conditions.
The particular integral represents the steady-state response to the driving force and is given by xp(t) = Pcos(ωt - δ), where P is the amplitude of the forced oscillation and δ is the phase difference between the displacement and the driving force.

Steady State Behavior

The oscillator oscillates with the same frequency as the periodic force, known as the driving force.
If the natural frequency (ω0) and the driving frequency (ω) are very close, beats will be produced.
The duration of these beats is determined by the damping coefficient β.
The phase difference between the oscillator and the driving force is δ = tan⁻¹(2βω/(ω0² - ω²)).
The amplitude of the driven oscillator in the steady state is given by A = (f0 / √((ω0² - ω²)² + 4β²ω²)).
The amplitude depends on the difference between the natural frequency squared and the driving frequency squared.
When the driving force frequency is much larger than the natural frequency (ω >> ω0) and damping is small (β ≈ 0), the amplitude is inversely proportional to the mass of the oscillator, indicating mass-controlled motion.

Diffraction

Diffraction occurs when waves encounter an obstacle or a narrow opening comparable in size to the wavelength of the wave.
This phenomenon results in the bending of the waves, which is most noticeable when the obstacle is in the path of the wave.

Types of Diffraction

Fresnel Diffraction: Happens when the distance of either the source or the screen or both from the obstacle is finite.
Fraunhofer Diffraction: Happens when the distance of either the source or the screen or both from the obstacle is infinite.
Fraunhofer Diffraction can be achieved using lenses to make the incident wavefront plane.

Fraunhofer Diffraction due to a single slit

A parallel beam of monochromatic light incident on a single slit AB of width 'e' produces a diffraction pattern on a screen.
The rays of light passing through the lens L2 at an angle θ converge at a point P1 on the screen, forming an image.
The path difference between the rays reaching P1 is BK = ABsinθ = esinθ.
The phase difference between the rays is given by (2π/λ) * esinθ.

Polarization

Linearly Polarized/Plane Polarized: The vibrations of the light are confined to a single linear direction at right angles to the direction of propagation.
Circularly Polarized: Two plane polarized waves superimpose, resulting in the resultant light vector rotating with a constant magnitude in a plane perpendicular to the direction of propagation and tracing a circle.
Elliptically Polarized: Two plane polarized waves superimpose, resulting in the resultant light vector varying periodically in magnitude during its rotation, tracing an ellipse.

Quarter Wave Plate

A double refracting crystal plate with a thickness that produces a path difference of λ/4, or a phase difference of π/2 between the ordinary and extraordinary waves.
It is constructed by cutting a plane from a double refracting crystal with its face parallel to the optic axis.
When a beam of light is incident on the plate, it gets divided into the ordinary (O) ray and the extraordinary (E) ray.
The path difference between these rays is (μO - μE)t, where μO and μE are the refractive index of the O-ray and E-ray, respectively, and t is the thickness of the plate.
For the plate to act as a quarter wave plate, (μO - μE)t = λ/4.
The thickness "t" can be calculated as t = λ / [4 (μO - μE)] for positive crystals and t = λ / [4 (μE - μO)] for negative crystals.

Uses of Quarter Wave Plate

Produces circularly and elliptically polarized light.
Used with Nicol prisms for analyzing various types of polarized light.

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Description

Explore the basics of L.C circuits, where an inductor and capacitor work together in a DC circuit. Understand how the capacitor charges and discharges, the relationship between charge and current, and the energy dynamics in these circuits. This quiz will test your knowledge of electrical principles governing L.C circuits.