Voltage Source across Resistor

 

V(t) = V_m sin ωt

i(t) = I_m sin ωt

And P(t) = v(t) i(t)

P(t) = V_m sin ωt I_m sin ωt

Hence, average power is

It is clear that if current or voltage waveform has a frequency of 50 Hz then power waveform has a frequency of 100Hz.

Voltage Source across Inductor

V(t) = ωL I_m cos ωt

V(t) = X_LI_m cos ωt

Where; X_L = ωL; X_L is known as inductance of circuit and has a unit of “ohm”.

V(t) = XLI_m sin (ωt + 90⁰)

And instantaneous power

p(t) = v (t) * i(t)

= V_m cos ωt I_m sin ωt

So average power is 

Hence in positive half cycle of the power, inductor takes energy from the source and in the negative cycle inductor delivers energy to the source. Hence not power dissipation is zero.

From above mathematical expressions we can draw phasor diagram of rms value of voltage and current.

From vector diagram it is clear that in an inductor current lags to voltage by  (or voltage leads to currents by 90⁰).

Uniform circular motion

If the particle moves in the circle with a uniform speed, we call it a uniform circular motion. In this case, dv/dt = 0 and equation gives .

Thus, the acceleration of the particle is in the direction of , that is, towards the centre. The magnitude of the acceleration is α = ω²r.



Thus, if a particle moves in a circle of radius r with a constant speed v, its acceleration is v²/r directed towards the centre. This acceleration is called centripetal acceleration. Note that the speed remains constant, the direction continuously changes and hence the “velocity” changes and there is an acceleration during the motion.

The linear acceleration is .

This acceleration is directed towards the centre of the circle.

Non uniform circular motion: If the speed of the particle moving in a circle is not constant, the acceleration has both the radial and the tangential components. According to equation, the radial and the tangential acceleration are 

Thus, the component of the acceleration towards the centre is ω²r = - v²/r and the component along the tangent (along the direction of motion) is dv/dt. The magnitude of the acceleration is .

The direction of this resultant acceleration makes an angle α with the radius where .

Total Internal Reflection

When light travels from an optically denser medium to a rarer medium at the interface, it is partly reflected back in to the same medium and partly refracted to the second medium. This reflection is called the internal reflection.

When a ray of light enters from a denser medium to a rarer medium. It bends away from the normal, for example, the ray AO₁B in figure. The incident ray AO₁ is partially reflected (O₁C) and partially transmitted (O₁B) or reflected, the angle of refraction (r) being larger than the angle of incidence (i) As the angle of incidence increases, so does the angle of refraction, till for the ray AO₃. The angle of refraction is π/2. The refracted ray is bent so much away from the normal that it grazes the surface at the interface between the two media. This is shown by the ray AO₃D in figure. If the angle of incidence is increased still further (e.g., the ray AO₄), refraction is not possible, and the incident ray is totally reflected.

This is called total internal reflection, when light gets reflected by a surface, normally some fraction of it gets transmitted. The reflected ray, therefore, is always less intense than the incident ray, howsoever smooth the reflecting surface may be. In total internal reflection, on the other hand, no transmission of light takes place.

The angle of incidence corresponding to an angle of refraction 90⁰ say ∠AO₃N is called the critical angle (i_c) for the given pair of media. We see from Snell’s law  that if the relative refractive index is less than one then, since the maximum value of  is unity, there is an upper limit to the value of sin i for which the law can be satisfied, that is, i = i_c such that sin i_c = n₂₁.

Source across Capacitor

Where,  and is known as capacitive reactance

i (t) = I_m sin (ωt + 90⁰)

Instantaneous power

p(t) = v(t) i(t)

= V_m sinωt I_m cos ωt

Average power

Now vector diagrams are drawn below:

Hence in a capacitor current leads to voltage by 90° (or voltage lags to currents by90°).

Series LCR Circuit

The applied voltage V divides into three parts, V_L (across L), V_C (across C) and V_R (across R) such

that

We know

By KVL, V = V_R + j (V_L - V_C)

V = √ (V²_R + (V_L - V_C)²)

V_L = Voltage across Inductor

V_R = Voltage across Resistor

V_C = Voltage across Capacitor

The impedance of the circuit is

V_R = IR; V_C = ; V_L = IωL;

V = IZ

But at some particular frequency both Inductive effect and Capacitor effect cancels each other and the circuit start oscillating. This is called resonance and the resonating frequency is

At this frequency the current in the circuit is maximum as impedance (Z) is minimum.

RMS Value & Average Value

RMS Value:

RMS value is defined on the basis of the heating effect of the waveform. The ac voltage at which heat produced in an ac circuit is equal to heat produced in the dc circuit is called as V_rms, provided both ac circuit ac circuit and dc circuit have equal value of resistance and operated for same time period.

P_dc = I²R

W_dc = I²Rt

And P_ac = i²R

So, W_ac = i²Rt

As per definition W_ac = W_dc

The general formula for finding the rms value is,

Or for sinusoidal waveform

Here we shall now derive rms value for some waveforms which are commonly encountered.

Average Value:

Average value is defined on the basis of the charge transfer in the circuit. The ac voltage, at which the charge transfer in ac circuit is equal to charge transfer dc circuit, is called as V_avg provided both ac and dc circuits are operated for same time period and having equal value of resistance.

I = V/R; i = v/R

I = Q/t, Q_ac = i x t

Q_dc = I x t

As per definition,

Q_dc = Q_ac

The average value can be calculated as, V_avg = 1/T ^T∫₀ V(t) dt

Resistors in Series and Parallel

Sometimes we are interested to replace a relatively complicated resistor combinations with a single equivalent resistor, especially when we are not interested in the current, Voltage or power associated with any of the individual resistors in the combinations. All the current, voltage and power relationships in the remainder of the circuit will be unchanged.

Consider the series combinations of N - resistors.

First apply KVL,

v_s = v₁ + v₂ + … + v_N

And then ohm’s law,

v_s = R₁i+ R₂i + … + R_Ni

= (R₁ + R₂ + … + R_N) i

Now compare this result with the simple equation applying to the equivalent circuit.

v_s = R_eqi

Thus, the value of the equivalent resistance for N - series resistors is

R_eq = R₁ + R₂ + … + R_N

Similar simplification can be applied to parallel circuits. A circuit containing N - resistors in parallel, as in leads to KCL equation,

i_s = i₁ + i₂ + … + i_N

Or 

Thus, 

Photo Electric Effect

It is the emission of electrons from the surface of certain substances, mainly metals, when they are illuminated by electromagnetic radiations like X-rays, ultraviolet and even visible light. The electrons emitted are called photo electrons.

Experiment to Demonstrate Photoelectric Effect:

In figure (a) A and C are two zinc plates, connected to positive and negative of a battery E respectively, through a micro-ammeter M and a key K. The plates A and C are enclosed in an evacuated quartz bulb. The quartz will not absorb ultraviolet rays. It is evacuated, so that, the metal surface remains pure.

The micro ammeter M registers a current, whenever ultraviolet rays fall on the zinc plate C. These electrons travel towards the plate A and thus a current flows through the circuit. If ultraviolet rays are stopped, there is no deflection in the micro ammeter. This means, electrons are ejected from the plate C by the radiations falling on it.

Laws of Photo Electric Emission:

⇒ For a given photosensitive material, there is a minimum frequency below which there is no photoelectric emission. This frequency is independent of the intensity of light and is called the threshold frequency.

⇒ Photoelectric emission is an instantaneous phenomenon. There is no time lag between the incidence of radiation and the emission of photo electrons.

⇒ The number of photo electrons and the photoelectric current is directly proportional to the intensity of incident radiation (provided the frequency  is greater than the threshold frequency).

⇒ The kinetic energy of photo electrons increases with the frequency of the incident radiation and is independent of the intensity of radiation.