Physics Test - 10

According to Brewster's law of polarization, where is angle of incidence

KEY CONCEPTS

OTHER CONCEPT(S)

Polarization by Reflection (Brewster's Law)

Polarisation by reflection : Brewster discovered that when a beam of unpolarised light is reflected is from a transparent medium (refractive index =μ), the reflected light is completely plane polaised at certain angle of incidence (called the angle of polarisation θ_p ).

iii Electric field of polarised reflected light will oscillate in a plane perpendicular to plane of incidence.

iv. component of electric field in the plane of reflection get absorbed and will not contribute in reflected light but component of electric field parallel to plane of reflection is divided into reflected and refracted ray so reflected light will have intensity less than half of incident unpolarised light

Polarization by Refraction

By double refraction : In certain crystals, like calcite, quartz and tourmaline etc, incident unpolarized light beams of equal intensities with perpendicular polarization.

(i) One of the ray is ordinary ray (O-ray) it obey's the Snell's law. Another ray's extra ordinary ray (E-ray) it doesn't obey's the Snell's law.

(ii) Along a particular direction (fixed in the crystal, the two velocities (velocity of O-ray V₀ and velocity of E-ray V_e) are equal; this direction is known as the optic axis of the crystal (crystal's known as uniaxial crystal). Optic axis is a direction and not any line in crystal

KEY CONCEPTS

Work Function In Photoelectric Effect
Einstein believed that to give a single electron the energy to move, the metal was hit by a single photon (destroying itself), and transferred its energy to the one electron.

$•$ Since the electron is originally attached to the metal, some minimum amount of energy must be needed just to snap it off. Otherwise, electrons would just be dropping off of atoms all the time.
$•$Einstein called this the work function of the metal, since you needed to do work on the electron to break it off.
Every metal has its own work function, since different metals hold on to their electrons with different strengths.
$•$Popping the electrons off starts to happen at a minimum threshold frequency, so that must correspond to the work function.
The formula for this is a modification of Planck's formula

W = work function (J)
h = Planck's Constant

f₀ = threshold frequency (Hz) = c(speed of light) / threshold wavelength (m)

Einstein's Equation of Photoelectric effect
Einstein's equation can be written as

Critical angleθc=sin^-1 (⁡1/μ)

Wavelength increases in the sequence of VIBGYOR. According to Cauchy's formula refractive index( μ )decreases as the wavelength increases. Hence the refractive index will increase in the sequence of ROYGBIV.

The critical angleθ_Cwill thus increase in the same order VIBGYOR. For green light the incidence angle is just equal to the critical angle. For yellow, orange and red the critical angle will be greater than the incidence angle. So these colours will emerge from the glass air interface. Hence, the correct option is (Yellow, orange, red).

KEY CONCEPTS

Total internal Reflection
Total internal reflection is a phenomenon which occurs when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflection occurs. This is particularly common as an optical phenomenon, where light waves are involved, but it occurs with many types of waves, such as electromagnetic waves in general or sound waves.

To solve the problem, we need to analyze the relationship between the lengths of the brass and steel rods when they are subjected to temperature changes. We are given that the difference in lengths of the two rods, (l₂−l₁), remains constant at all temperatures.

Power
In physics, power is the rate of doing work. It is equivalent to an amount of energy consumed per unit time. In the SI system, the unit of power is the joule per second (J/s), known as the watt in honour of James Watt, the eighteenth-century developer of the steam engine.

As a simple example, burning a kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more quickly, it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration ∆t, the average power P_avg over that period is given by the formula

It is the average amount of work done or energy converted per unit of time. The average power is often simply called "power" when the context makes it clear.

The instantaneous power is then the limiting value of the average power as the time interval ∆t approaches zero.

C-14 is the element used in radioactive carbon dating

KEY CONCEPTS

Radioactivity

The phenomenon of spontaneous emission of radiatons by heavy elements is called radioactivity. The elements which shows this phenomenon are called radioactive elements.Radioactivity is a nuclear event and not atomic. Hence electronic configuration of atom don't have any relationship with radioactivity.

Radioactive Decay Law

dN=−λNdt        Activity: dN/dt=−λN

Activity: It is defined as the rate of disintegration (or count rate) of the substance (or the number of atoms of any material decaying per second) i.e.

whereA₀ = Activity of t = 0, A = Activity after time t

Units of activity (Radioactivity)

It's units are Becqueral (Bq), Curie (Ci) and Rutherford (Rd)

1 Becquerel = 1 disintegration/sec,
A=A₀e^−t

A = Activity remaining in the radioactive material after time (t) from when the initial activity was measured (at time of purchase)

Ao= Initial Activity of the radioactive material (time of purchase)

KEY CONCEPTS

RLC circuit

Introduction: An RLC circuit (or LCR circuit or CRL circuit or RCL circuit) is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. The circuit forms a harmonic oscillator for current and will resonate in a similar way as an LC circuit will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source. This effect of the resistor is called damping. The presence of the resistance also reduces the peak resonant frequency somewhat. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a component. An ideal, pure LC circuit is an abstraction for the purpose of theory. Resonance: resonance frequency, f0, angular frequency,

Resonance
An important property of this circuit is its ability to resonate at a specific frequency, the resonance frequency, ∫₀. Frequencies are measured in units of hertz. In this article, however, angular frequency, ω₀, is used which is more mathematically convenient. This is measured in radians per second. They are related to each other by a simple proportion,
ω₀ = 2π∫₀

Generating AC

AC can be produced using a device called an alternator. This device is a special type of electrical generator designed to produce alternating current.

A loop of wire is spun inside of a magnetic field, which induces a current along the wire. The rotation of the wire can come from any number of means: a wind turbine, a steam turbine, flowing water, and so on. Because the wire spins and enters a different magnetic polarity periodically, the voltage and current alternates on the wire.

Waveforms

AC can come in a number of forms, as long as the voltage and current are alternating. If we hook up an oscilloscope to a circuit with AC and plot its voltage over time, we might see a number of different waveforms. The most common type of AC is the sine wave. The AC in most homes and offices have an oscillating voltage that produces a sine wave.

Triangle waves are found in sound synthesis and are useful for testing linear electronics like amplifiers.

Phase relation between voltage and current

Physical quantity which represents both the instantaneous value and direction of alternating quantity at any instant is called it's phase. It's a dimensionless quantity and it's unit is radian.

If an alternating quantity is expressed as X=X₀sin (ωt±ϕ₀) then the argument of sin(ωt+ϕ) sinωt+ϕ is called it's phase. Where ω t = instantaneous phase (changes with time) and ϕ₀ = initial phase (constant w.r.t. time)

Phase difference (Phase constant) : The difference between the phases of currents and voltage is called phase difference. If alternating voltage and current are given by V=V₀ sin(ωt+ϕ₁) and i=i₀ sin(ωt+ϕ₂) then phase difference ϕ=ϕ₁−ϕ₂ (relative to current) or ϕ=ϕ₂−ϕ₁ (relative to voltage)

Examples on Phase difference, Voltage difference of circuit elements

Example. A 9/100π H inductor and a 12 ohm resistance are connected in series to a 225V, 50Hz ac source. Calculate the current in the circuit and the phase angle between the current and the source voltage.

Introduction: An RLC circuit (or LCR circuit or CRL circuit or RCL circuit) is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. The circuit forms a harmonic oscillator for current and will resonate in a similar way as an LC circuit will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source. This effect of the resistor is called damping. The presence of the resistance also reduces the peak resonant frequency somewhat. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a component. An ideal, pure LC circuit is an abstraction for the purpose of theory. Resonance: resonance frequency, f0, angular frequency,

Resonance

An important property of this circuit is its ability to resonate at a specific frequency, the resonance frequency, ∫0. Frequencies are measured in units of hertz. In this article, however, angular frequency, ω_0, is used which is more mathematically convenient. This is measured in radians per second. They are related to each other by a simple proportion,

ω₀ = 2π∫₀

Natural frequency

Mass of on electron moving with speed \({v}\) is \(\frac{{m}_{{o}}}{\sqrt{1-\frac{{v}^{2}}{\mathrm{c}^{2}}}}\).

Where, \({m}_{o}\) is rest mass

\({v}\) is speed of electron

\({c}\) is the speed of light

So, \({m}=\frac{{m}_{{o}}}{\sqrt{{1 - \frac { { v } ^ { 2 } } { { c } ^ { 2 } }}}}\)

\(=\frac{{m}_{{o}}}{\sqrt{1-\frac{\left(\frac{\sqrt{3} {c}}{2}\right)^{2}}{{c}^{2}}}}\)

\(=\frac{{m}_{{o}}}{\sqrt{1-\frac{3}{4}}}\)

\(=\frac{{m}_{{o}}}{\sqrt{\frac{1}{4}}}\)

\(=\frac{{m}_{{o}}}{\frac{1}{2}}\)

\(=2 {~m}_{{o}}\)

So, the mass becomes \(2\) times of its rest mass.

From Stefan's law, the rate at which energy is radiated by sun at its surface is P = σ × 4πR²T⁴

[Sun is a perfectly black body as it emits radiations of all wavelengths and so for it e=1]

The intensity of this power at earth's surface is

The area of earth which receives this energy is only one-half of total surface area of earth, whose projection would be πr₀²

KEY CONCEPTS

Stefan's law
According to it the radiant energy emitted by a perfectly black body per unit area per sec (i.e. emissive power of black body) is directly proportional to the fourth power of its absolute temperature, i.e.

One end of a rod of length 20 cm is inserted in a furnace at 800 K. The sides of the rod are covered with an insulating material and the other end emits radiation like a black body. The temperature of this end is 750 K in the steady state. The temperature of the surrounding air is 300 K. Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conductivity of the rod. Stefan constant σ=6.0×10⁻⁸W/m²k⁴.