ALTERNATING CURRENT

JEE MAIN CHAPTER OVERVIEW

Chapter Priority	3 (LOW)
Number of problems in PYQ	HIGH
Difficulty	HIGH/MODERATE

To solve JEE MAIN problems for AC, you need to remember all formulae given in these notes.

Note: Questions can be tricky especially in RLC circuit.

TYPES OF CURRENT

• DIRECT CURRENT: Current that does not change direction with constant magnitude

• PULSATING CURRENT: Pulsating direct current is a periodic current which changes in value but never changes direction.

• ALTERNATING CURRENT: Current is said to be alternating if it is change continously in magnitude and perodically in direction with time.

CONDITION REQUIRED FOR CURRENT/VOLTAGE TO BE ALTERNATING

• Amplitude is constant

• Alternate half cycle positive and negative

VARIOUS TYPES OF ALTERNATING CURRENT

We will study Sinusoidal AC for JEE MAIN and JEE ADVANCED

SINUSOIDAL AC

Current varies according to sin or cos function.

In the above figure, current is varying with time according to the function:

Key terms in the function:

I0 = Amplitude of current

w (omega) = Angular frequency

wt = Phase = angle inside the sine function

TIME PERIOD(T): The time taken by alternating current to complete one cycle of variation. It is measured in seconds.

FREQUENCY(f): The number of cycle completed by an alternating current in one second. It is measured in Hz (Hertz)

RELATION BETWEEN TIME PERIOD, FREQUENCY & ANGULAR FREQUENCY

Current can be represented both as a sine function or cosine function.

AVERAGE VALUE OR MEAN VALUE

Average value is define over a period of time.

For any time period, average value is the total charge flown divided by the time.

General formula for average current over time t is given by:

For a complete cycle (where time is equal to time period T), for a sinusoidal AC, average value of current is ZERO (0).

For a positive half cycle (where time equals half the time period T/2), for a sinusoidal AC, average value of current is given by:

REMEMBER the following for average values of AC:

SPECIAL NOTE ON AVERAGE VALUES for sin and cos:

ROOT MEAN SQUARE VALUE

For Alternating Current, root mean square value over a time (t) is defined as:

[Note: Formula given above is the general formula to calculate RMS value for any quantity, not just current. Simply replace Current with the quantity you wish to get RMS value for. ]

RMS value of current is related to heat produced in the circuit.

RMS Current equals the value of the Direct Current which will produce the same amount of heat in the circuit as the given Alternating Current in the same amount of time.

Note: RMS value is same for both half (T/2) and full cycle (T)

For sinusoidal AC, RMS Value (0.707Io) is greater than Average Value over half cycle(0.637Io).

Note (Important): If nothing is mentioned in the question, then values printed in a.c circuit on electrical appliances, reading of AC meters etc. are assumed to be RMS.

TRY A JEE MAIN PYQ

Q. The current flowing through an ac circuit is given by I = 5sin (120πt) A. How long will the current take to reach the peak value starting from zero? [JEE MAIN 2022]

[If you cannot solve any doubts related to any chapter of physics, upload a photo of your doubt at www.savai.co.in or WhatsApp me @ 7982286138, we will send solution.]

Q. A resistance of 40 Ω is connected to a source of alternating currentrated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to rms value.

[JEE MAIN 2022]

[If you cannot solve any doubts related to any chapter of physics, upload a photo of your doubt at www.savai.co.in or WhatsApp me @ 7982286138, we will send solution.]

Q. An alternating current is given by the equation i = i1sinωt + i2cosωt. The rms current will be: [JEE 2021]

Find solution below:

[If you cannot solve any doubts related to any chapter of physics, upload a photo of your doubt at www.savai.co.in or WhatsApp me @ 7982286138, we will send solution.]

MEASUREMENT OF AC

Alternating current and voltages are measured by a.c. ammeter and a.c. voltmeter respectively. Working of these instruments is based on heating effect of current, hence they are also called hot wire instruments.

Note: AC ammeter and AC voltmeter measure RMS values (i.e., they measure heat produced and then calculate RMS values)

Reading of DC meter when attached to AC circuit will be ZERO.

PHASE, PHASE DIFFERENCE & PHASORS

Consider the equation for AC current:

Here,

𝜙 = initial phase (phase at t = 0, always constant)

wt+𝜙 = instantaneous phase (phase at any time 't', varies with 't')

Instantaneous phase (or just phase) decides both magnitude and sign of current.

PHASE DIFFERENCE

Let's say Voltage varies as:

Current varies as:

Then at any time the phase difference of V w.r.t I is (𝜙1 - 𝜙2)

If 𝜙1 > 𝜙2, we say that Voltage leads or Current lags, i.e., Voltage reaches maximum value before Current If 𝜙1 < 𝜙2, we say that Voltage lags or Current leadss, i.e., Current reaches maximum value before Voltage

PHASOR DIAGRAM

• It is just a technique used to solve trigonometric equations.

• We can represent voltage and current in the circuit using Phasors.

• All voltages and currents in a single phasor diagram must be of the same frequency

• A phasor is a vector whose one end is fixed at the origin.

• Its magnitude represents the maximum value of the quantity it represents

• The angle the phasor makes with the x-axis represents the phase.

[Note: Phasors are a representation only. Voltage and Current are scalars and not vectors. We only use Phasors to solve questions faster.]

Given below is a Phasor diagram.

AC CIRCUIT CONTAINING PURE RESISTANCE

AC CIRCUIT CONTAINING PURE CAPACITANCE

In a purely capacitive circuit, current leads the voltage by a phase of 90 degrees.

Maximum current is the ratio of Maximum voltage divided by the reactance (Xc)

Xc = 1/(wC)

where w = angular frequency and C is the capacitance.

AC CIRCUIT CONTAINING PURE INDUCTANCE

In a purely inductive circuit, current lags the voltage by a phase of 90 degrees.

Maximum current is the ratio of Maximum voltage divided by the reactance (XL)

XL = wL

where w = angular frequency and L is the inductance.

RLC Circuit in SERIES

Most of the difficult questions in JEE MAIN AND ADVANCED from this topic are from RLC circuits. First let us understand RLC circuits in series and then we will briefly discuss RLC circuits in parallel.

In a series RLC circuit with a Voltage source e = eosin(wt), current in the circuit may lag or lead the voltage depending upon the values of capacitance and inductance in the circuit.

The phase difference between I & V is given by:

tan(𝜙) = (Xc-XL)/R

If Xc>XL, circuit has capacitive nature and current leads.

If Xc<XL, circuit has inductive nature and current lags.

The maximum current in the circuit (I0) is given by the ratio of maximum voltage and impedance (Z).

We can also look at an extended phasor diagram to understand RLC circuit better:

VC = Voltage across Capacitor whose max value is I0XC

VL = Voltage across Inductor whose max value is I0XL

VR = Voltage across Resistor whose max value is I0R

Remember,

Vc will lag Current by 90 degrees

VL will lead Current by 90 degrees

VR will be in phase of Current

Voltage across the circuit will be given by the vector sum of VR, VC, VL

Just like there is resistance, for AC circuits we define a term called Z (impedance)

The relationship between impedance (Z), phase(𝜙), Resistance (R), and reactance (Xc - XL) can be shown using the impedance diagram:

If XC > XL, 𝜙 is positive and the circuit is predominantly capacitive. Consequently, the current in the circuit leads the source voltage. If XC < XL, 𝜙 is negative and the circuit is predominantly inductive. The current in the circuit lags the source voltage.

Power in AC Circuit: The Power Factor

Power dissipated in a general AC circuit is:

P = Vrms*Irms*cos(𝜙)

Case 1: Purely resistive circuit

𝜙 = 0, therefore, cos(𝜙) = 1, which means maximum power = Vrms*Irms is dissipated

Case 2: Purely inductive or capacitive circuit

𝜙 = +90 or -90 degrees, cos(𝜙) = 1, which means no power is dissipated on average.

Case 3: RLC circuit:

Depending upon the value of 𝜙, we will get power dissipated, in such cases, power is dissipated only through resistor

cos(𝜙) = POWER FACTOR = R/Z (very important)

Watt-less Current

Watt-less current in a circuit = Irms*sin(𝜙)

Sometimes questions are asked directly on this concept, it is the current that does not cause heat loss in the circuit. You will be asked to calculate the watt-less current.

TRY A JEE MAIN QUESTION

Q. What happens to the inductive reactance and the current in a purely inductive circuit if the frequency is halved? [JEE MAIN 2021]

[If you cannot solve any doubts related to any chapter of physics, upload a photo of your doubt at www.savai.co.in or WhatsApp me @ 7982286138, we will send solution.]

Q. In a series LCR circuit, the inductive reactance (XL) is 10 Ω and the capacitive reactance (XC) is 4 Ω The resistance (R) in the circuit is 6 Ω. The power factor of the circuit is:

[JEE MAIN 2021]

[If you cannot solve any doubts related to any chapter of physics, upload a photo of your doubt at www.savai.co.in or WhatsApp me @ 7982286138, we will send solution.]

RESONANCE

All RLC circuits have a natural frequency. If the frequency of the source and the natural frequency match, then amplitude if current in the circuit will be very high. This is called resonance.

Condition for resonance is given in terms of angular frequency:

QUALITY FACTOR

Q-factor of AC circuit basically gives an idea about stored energy & lost energy in a circuit.

Remember the following but don't go into details for JEE MAIN.

• Higher the quality factor, lower the angular bandwidth, more sharpness

• Lower the quality factor, higher the angular bandwidth, less sharpness

Given below is a graph between current amplitude and angular frequency.

Variation of Z with angular frequency

At resonant frequency, Z is minimum and equal to resistance

Brief Note on Parallel Circuits

[Rarely asked in MAIN, not common even in ADVANCED. Simply read so you are not surprised in exam if it comes.]

In such cases, Voltage across Resistor, Capacitor and Inductor is same. Current is different across each component.

In Resistor, current will be in phase with Voltage.

In Capacitor, current will lead the Voltage.

In Inductor, current will lag the Voltage.

Given below is the Phasor diagram for the parallel RLC circuit

Current in the circuit is given by the following equation:

IR = Max current through R

IC = Max current through C

IL = Max current through L

Another type of Parallel Circuit

Given below is the phasor diagram (skip if you don't understand, not very important for JEE MAIN)

• Voltage across C is the same as that provided by the source.

• Voltage across LR (combined) is the same as that provided by the source.

• Voltage across C is in phase with Voltage across LR (combined)

Consider them as two separate circuits:

• An LR circuit connected to an AC source.

• A purely capacitive circuit connected to an AC source.

• Currents in both circuits will be added like Phasors.

• Current in Capacitive circuit leads Voltage by 90 degrees.

• Current in LR circuit lags by some angle 𝜙 depending on the impedance of RL circuit.

LC OSCILLATIONS

All you need to know is:

• Charge in capacitor and current in inductor oscillate like an SHM. Charge is similar to position (x) and current is similar to velocity (v) of an SHM.

• Total Energy remains constant just like an SHM.

• Just like an SHM, there is a angular frequency of oscillation w(omega)

Given below is a derivation of LC oscillation. Just remember the formula for angular frequency (w) of oscillation & total energy.

Angular frequency of LC Oscillations is given by:

Total Energy of System is constant although it may be distributed across inductor and capacitor.

TRANSFORMER

It is used to change a voltage to another greater or smaller value. It uses mutual induction to achieve voltage transformation.

It comprises an iron core in which two coils are wound. One coil with Np turns is called primary coil (input). The other coil with Ns turns is called secondary coil (output).

Voltage in secondary (Vs) coil can be related to voltage in primary coil (Vp) as:

Current in secondary (Vs) coil can be related to current in primary coil (Vp) as:

Hope you can REVISE Alternating Current FOR JEE MAIN PHYSICS from this guide to answer all varieties of JEE MAIN PYQs. Thanks a lot!

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