Physics - AC Voltage Applied to an Inductor Concept Quick Start

© ScoreLab by Profsam.com Designed to help CBSE Class 12 students improve conceptual clarity and score up to 30% more marks in Physics, Chemistry, and Mathematics. Profsam.com Topic: AC Voltage Applied to an Inductor

Unit: Unit 7: Alternating Current

Class: CBSE CLASS XII

Subject: Physics

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SECTION 1: WHY THIS TOPIC MATTERS

Understanding how inductors behave with alternating current (AC) isn't just a topic for your exams; it's a cornerstone of the technology that powers our modern world. From the national power grid to the radio in your car, the principles of AC circuits are constantly at work. This concept explains how certain components can resist the change in electricity, a property that engineers use to build sophisticated and essential devices. Here are a few real -world applications that rely on this fundamental concept:

Power Systems: Large inductors are used in power grids to limit massive surges in

current during a fault (like a short circuit). Their ability to resist sudden changes acts as a crucial safety mechanism, protecting the electrical infrastructure.

Tuning Circuits: Whenever you tune a radio to a favourite station, you are taking

advantage of this principle. Inductors and capacitors create tuning circuits that respond strongly to one frequency while ignoring all others, letting you select a single broadcast. While the physics behind inductors might seem complex at first, a simple physical analogy can make their behavior feel intuitive and easy to grasp.

SECTION 2: THINK OF IT LIKE THIS

One of the best ways to understand a new physics concept is to connect it to a physical experience. A good mental model can make the behavior of an inductor feel intuitive, translating abstract mathematics into a tangible feeling of inertia or delay. The Water Pipe with Inertia (Heavy Liquid) Imagine trying to push a very heavy liquid, like mercury, back and forth through a pipe.

When you start pushing (applying pressure, which is like Voltage), the heavy liquid

doesn't start moving immediately. Its inertia resists the change in motion.

This resistance to acceleration creates a Delay.
Only after a moment does the liquid start to flow (the Current).

© ScoreLab by Profsam.com Designed to help CBSE Class 12 students improve conceptual clarity and score up to 30% more marks in Physics, Chemistry, and Mathematics. Profsam.com By the time the liquid is flowing at its maximum speed, you might have already eased off the pressure. This is precisely how an inductor works: the voltage (the push) happens before the current (the flow) can catch up.

Push (Voltage) → Delay (Inertia) → Flow (Current)

The Spinning Flywheel

Another excellent analogy is spinning a heavy flywheel. When you apply a force (voltage) to make it spin, its rotational inertia prevents it from speeding up instantly. There is a noticeable lag between when you apply the maximum force and when the flywhee l reaches its maximum speed (current). This, again, visualizes the core concept of inertia causing a delayed response. These physical analogies of "delay" have an exact mathematical description, which is the official definition required for your board exams.

SECTION 3: EXACT NCERT ANSWER (LEARN THIS FOR EXAMS)

This section contains the precise definitions and formulas from the NCERT textbook.

For scoring well in your exams, it is crucial to learn and reproduce these key points accurately. -------------------------------------------------------------------------------- NCERT Core Concepts: Pure Inductor Applied Voltage The AC voltage applied to a pure inductor is given by: v = v_m sin ωt Resulting Current The current that flows in the inductor as a result is: i = i_m sin( ωt − π/2) Amplitude of the Current The peak current i_m is determined by the peak voltage and the inductive reactance: i_m = v_m / ( ωL) Inductive Reactance The opposition to the flow of alternating current is called inductive reactance X_L: X_L = ωL Phase Relationship "the current lags the voltage by π/2 or one-quarter (1/4) cycle." Power Consumption "the average power supplied to an inductor over one complete cycle is zero." Definition of Symbols

v: Instantaneous voltage (the voltage at any specific moment in time).
v_m: Peak (maximum) voltage.
ω (omega): Angular frequency of the AC source (measured in rad/s).
t: Time.

i: Instantaneous current (the current at any specific moment).
i_m: Peak (maximum) current.
π (pi): The mathematical constant (~3.14).
L: Self-inductance of the inductor (SI Unit: Henry, H).
X_L: Inductive reactance, the opposition to the change in current (SI Unit: ohm, Ω).

The next section will connect the "delay" we visualized in our water pipe analogy directly to the "− π/2" term in the official formula for current.

SECTION 4: CONNECTING THE IDEA TO THE FORMULA

This section bridges the gap between the intuitive physical behavior of an inductor and the mathematical formula provided by NCERT. Understanding this connection will help you remember the formula logically, not just by rote memorization. Here is the logical flow: 1. Fundamental Physics (Faraday's Law): The behavior of an inductor is governed by Faraday's Law of Induction.

For an inductor, this law takes the form v = L(di/dt) . This equation is the key: it states that the voltage across an inductor depends not on the current itself, but on the rate of change of the current ( di/dt). 2. Applying a Sinusoidal Current: Let's assume a sinusoidal current i(t) = i_m sin( ωt) is flowing through the inductor. To find the voltage, we need to find its derivative. 3.

The Derivative Connection: The derivative of a sine function is a cosine function. So, when we calculate di/dt, the voltage v(t) will be proportional to cos(ωt). 4. The Resulting Phase Shift: A cosine wave is mathematically identical to a sine wave that has been shifted forward by 90 degrees (or π/2 radians). That is, cos(ωt) = sin(ωt + 90°). 5.

The Final Form: This means the voltage v(t) follows a sin(ωt + 90°) pattern, while the current i(t) follows a sin(ωt) pattern. The voltage leads the current by 90°. Saying "voltage leads current by 90°" is exactly the same as saying " current lags voltage by 90°"—which perfectly matches the NCERT formula i = i_m sin( ωt − π/2).

In summary, the inductor's fundamental property of opposing change is what mathematically forces the current to lag behind the voltage by a perfect 90 -degree phase angle.

SECTION 5: STEP -BY-STEP UNDERSTANDING

Let's break down the entire process into a simple, sequential flow of ideas. This is perfect for building a clear mental picture and for quick revision.

An AC voltage is applied across the inductor, which tries to push current through it.

Due to its self -inductance, the inductor generates a back EMF that opposes any

change in the current flowing through it (Faraday's Law).

This opposition to change causes a delay. The current cannot build up instantly when

the voltage is applied.

The voltage reaches its peak value first. By the time the current overcomes the

inductor's inertia and reaches its own peak, the voltage has already completed a quarter of its cycle and started to decrease.

This delay is precisely a quarter of a cycle , which mathematically corresponds to a

90° or π/2 phase lag for the current relative to the voltage.

Over a full cycle, the inductor first stores energy in its magnetic field and then returns

that energy completely to the source. This is why the average power consumption is zero. Seeing these steps applied to a simple calculation will make the concept crystal clear.

SECTION 6: VERY SIMPLE EXAMPLE (TINY NUMBERS)

A numerical example helps solidify the relationship between voltage, reactance, and current.

Problem Statement

A pure inductor of 2 H is connected to an AC source given by v = 10 sin(5t) . Find the inductive reactance and the peak current. Step-by-Step Calculation Step 1: Identify Given Values From the voltage equation v = 10 sin(5t) , we can compare it to the standard form v = v_m sin( ωt):

Peak Voltage v_m = 10 V
Angular Frequency ω = 5 rad/s
Inductance L = 2 H

Step 2: Calculate Inductive Reactance (X_L) The formula for inductive reactance is X_L = ωL.

Calculation: X_L = 5 rad/s × 2 H = 10 Ω
The inductor provides 10 ohms of opposition to the changing current.

Step 3: Calculate Peak Current (i_m) The relationship is similar to Ohm's Law, i_m = v_m / X_L.

Calculation: i_m = 10 V / 10 Ω = 1 A

The maximum current that will flow through the circuit is 1 Ampere.

As you can see, the inductor's opposition (reactance) directly limits the peak current. Knowing the right formula is only half the battle; avoiding common mistakes is just as important.

SECTION 7: COMMON MISTAKES TO AVOID

Many students lose marks due to a few common, recurring misconceptions about inductors. This section will help you avoid those traps.

WRONG IDEA: Current and voltage are 180° out of phase (completely opposite).
Why students believe it: They hear "inductor opposes current" and think of it as a

direct, head -on fight, which suggests a 180° opposition.

CORRECT IDEA: Current and voltage are 90° out of phase (perpendicular). The

inductor opposes the change in current, not the current itself. This is a direct result of the voltage depending on the rate of change of current ( v = L di/dt ), which mathematically creates the quarter -cycle shift. --------------------------------------------------------------------------------

WRONG IDEA: Since an inductor stores and returns energy, it must have a negative

average power.

Why students believe it: They correctly understand that the inductor returns energy

to the source but incorrectly conclude this means the net power must be negative.

CORRECT IDEA: The energy borrowed from the source in one part of the cycle is

perfectly returned in another part. The instantaneous power is positive when the magnetic field is building and negative when it is collapsing. Over a full cycle, these positive and negative phases cancel out perfectly, making the average power exactly zero, not negative. A simple mnemonic can help lock in the correct phase relationship permanently.

SECTION 8: EASY WAY TO REMEMBER

Memory aids can be powerful tools to quickly recall key facts during a high -pressure exam situation. For AC circuits, one of the most famous mnemonics is: "ELI the ICEman" Here is how to use it:

In an inductor ( L), the Voltage ( E for EMF) comes before the Current ( I).
So, for an inductor, just remember ELI.

© ScoreLab by Profsam.com Designed to help CBSE Class 12 students improve conceptual clarity and score up to 30% more marks in Physics, Chemistry, and Mathematics. Profsam.com With the core ideas, analogies, and memory aids in place, it's time to consolidate the most important points for quick revision.

SECTION 9: QUICK REVISION POINTS

This section is your final checklist of the most important facts about an AC voltage applied to a pure inductor.

In a pure inductor, the current lags the voltage by exactly 90° (or π/2 radians).
The opposition to the flow of alternating current is called inductive reactance (X_L) ,

and its formula is X_L = ωL.

Inductive reactance is not constant ; it increases directly with the frequency of the AC

source. Higher frequency means more opposition.

The relationship between peak voltage and peak current follows an Ohm's Law -like

form: v_m = i_m × X_L .

A pure inductor does not consume energy . The average power dissipated over a

complete cycle is zero.

Energy is temporarily stored in the inductor's magnetic field during one part of the

cycle and is fully returned to the source in another part. For students who want to deepen their intuition, the final section offers some richer insights that connect this topic to other areas of physics and technology.

SECTION 10: ADVANCED LEARNING (OPTIONAL)

These points go beyond the basic exam requirements and are for students who want to build a deeper, more connected understanding of the topic.

Real-World Inertia: The spark you sometimes see when switching off an old fan,

motor, or other inductive load is the inductor's "inertia" in action. The inductor tries to keep the current flowing even after the circuit is broken, creating a large voltage spike across the swi tch contacts that is visible as a spark.

AC Motors: Most AC motors (like those in fans and pumps) are inductive loads. This

means they naturally cause the current to lag the voltage in the power grid. Power companies must manage this effect to ensure efficient energy delivery.

The "Why" Behind Zero Power: While current flows, the instantaneous power p(t) =

v(t)i(t) is positive when the source is delivering energy to build the magnetic field. It's negative when the magnetic field collapses and returns that energy to the source. The integral of this power over one full cycle is zero, meaning the average power is zero.

Connection to Faraday's Law: The entire 90° phase shift is not an arbitrary rule; it is a

direct mathematical consequence of Faraday's Law of Induction ( v = L di/dt ). This links © ScoreLab by Profsam.com Designed to help CBSE Class 12 students improve conceptual clarity and score up to 30% more marks in Physics, Chemistry, and Mathematics. Profsam.com the behavior of AC circuits directly back to the fundamental principles of electromagnetism you studied in a previous unit.

A Preview of Resonance: An inductor causes the voltage to lead the current by 90°

(+90° phase shift for voltage). You will soon learn that a capacitor does the exact opposite. This opposing behavior is the key to a phenomenon called resonance , where an inductor and capacitor in the same circuit can cancel each other's effects out, a principle used in every radio and TV tuner.