A coil of $0.01\ \text{H}$ inductance and $1\ \Omega$ resistance is connected to $200\ \text{V}$, $50\ \text{Hz}$ ac supply. Find the impedance of the circuit and time lag between maximum alternating voltage and current.

A coil of 0.01H inductance and 1Ω resistance is connected to 200V, 50Hz ac supply. Find the impedance of the c

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0.01\ \text{H}

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Concept understanding — Inductive Reactance Change

Inductive Reactance Change – A First Look

Imagine you're pushing a child on a swing. If you push at just the right moment — when the swing is coming back toward you — each push adds energy and the swing goes higher. But if you push at random moments, sometimes you push against the swing's motion, and it barely moves. The swing "resists" being pushed at the wrong time.

An inductor in an AC circuit behaves exactly like that swing. It doesn't resist current the way a resistor does (by turning energy into heat). Instead, it resists changes in current — and the faster the current tries to change, the more the inductor pushes back.

The Core Intuition

An inductor is just a coil of wire. When current flows through it, it creates a magnetic field. If the current tries to change — say, increase or decrease — the magnetic field changes too. That changing field induces a voltage in the coil that opposes the change in current. This is Lenz's law in action: the induced voltage always fights the change that caused it.

So the inductor acts like a kind of "inertia" for current. The more rapidly the current tries to change, the stronger the opposition. In a DC circuit, once the current settles to a steady value, the inductor stops opposing — it becomes just a wire. But in an AC circuit, the current is always changing direction, so the inductor is always fighting.

The Precise Statement

Inductive reactance ( $X_L$ ) is the opposition an inductor offers to alternating current. It depends on two things:

The inductance $L$ of the coil (measured in henries, H) — bigger coil, more opposition.
The frequency $f$ of the AC supply (measured in hertz, Hz) — faster changes, more opposition.

The formula is:

X_L = 2\pi f L

Where:

$X_L$ is in ohms ( $\Omega$ )
$f$ is the frequency in Hz
$L$ is the inductance in H

Key point: Unlike resistance, which is constant for a given resistor, inductive reactance changes with frequency. Double the frequency, double the reactance. Halve the frequency, halve the reactance.

What "Inductive Reactance Change" Means

When we talk about "inductive reactance change," we mean: how $X_L$ varies when either the frequency or the inductance changes.

Change	Effect on $X_L$	Why?
Frequency increases	$X_L$ increases	Current changes faster → stronger opposition
Frequency decreases	$X_L$ decreases	Current changes slower → weaker opposition
Inductance increases	$X_L$ increases	More magnetic field → more opposition
Inductance decreases	$X_L$ decreases	Less magnetic field → less opposition

Watch out

A common mistake is to think inductive reactance behaves like resistance. It doesn't. Resistance dissipates energy as heat; reactance stores and releases energy in the magnetic field. Also, reactance depends on frequency — resistance usually doesn't.

A Simple Example …

Why this formula?

Inductive Reactance Change: Why the Formula Holds

Let's build this from first principles — understanding why inductive reactance behaves as it does, not just memorizing $X_L = 2\pi f L$ .

1. The Core Idea: Opposition to Current Change

An inductor doesn't "resist" current like a resistor. Instead, it opposes changes in current due to self-induction.

When current changes, the magnetic flux through the inductor changes.
By Faraday's Law, a changing flux induces an emf (voltage) that opposes the change — this is Lenz's Law.
The induced voltage is proportional to the rate of change of current:

v_L = L \frac{di}{dt}

Where:

$v_L$ = induced voltage across inductor (V)
$L$ = inductance (henry, H)
$\frac{di}{dt}$ = rate of change of current (A/s)

2. Applying a Sinusoidal Current

In AC circuits, current is sinusoidal. Let:

i(t) = I_m \sin(\omega t)

Where:

$I_m$ = peak current (A)
$\omega = 2\pi f$ = angular frequency (rad/s)
$f$ = frequency (Hz)

Now compute the induced voltage:

v_L = L \frac{d}{dt}[I_m \sin(\omega t)] = L \cdot I_m \cdot \omega \cos(\omega t)

So:

v_L = \omega L \, I_m \cos(\omega t)

3. The Phase Shift: Voltage Leads Current

Notice:

Current: $\sin(\omega t)$
Voltage: $\cos(\omega t) = \sin(\omega t + 90^\circ)$

Voltage leads current by $90^\circ$ (or $\pi/2$ radians). This is a key property — the inductor causes a phase difference.

4. Defining Inductive Reactance

Reactance is the ratio of peak voltage to peak current (magnitude only, ignoring phase):

From above:

Peak voltage: $V_m = \omega L \, I_m$
Peak current: $I_m$

Thus:

X_L = \frac{V_m}{I_m} = \omega L

Since $\omega = 2\pi f$ :

\boxed{X_L = 2\pi f L}

Where $X_L$ is in ohms ( $\Omega$ ).

5. Why It Changes with Frequency

The formula reveals the why:

Higher frequency ( $f$ increases) → $\frac{di}{dt}$ is larger for the same current amplitude → larger induced voltage → greater opposition → $X_L$ increases. …

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The RL series circuit has impedance $Z=\sqrt{R^2+X_L^2}\approx3.30\ \Omega$ , and the current peaks a phase angle $\phi=\tan^{-1}(X_L/R)\approx72.3^\circ$ after the voltage, giving a time lag $\Delta t=\phi/\omega\approx4.02\text{ ms}$ .

An inductor opposes changes in current, so in this coil the current cannot follow the voltage instantly — it reaches its maximum a little after the voltage does. Two quantities are asked for: the impedance $Z$ and the time delay between the voltage peak and the current peak.

1. Inductive reactance

The coil's opposition to alternating current at $50\text{ Hz}$ is

X_L=2\pi f L=2\pi\times50\times0.01=\pi\approx3.14\ \Omega.

2. Impedance

Resistance $R$ and reactance $X_L$ are $90^\circ$ out of phase, so they combine like the legs of a right triangle:

Z=\sqrt{R^2+X_L^2}

Z=\sqrt{1^2+\pi^2}=\sqrt{1+9.87}=\sqrt{10.87}\approx3.30\ \Omega.

3. Phase angle

The current lags the voltage by an angle $\phi$ given by

\tan\phi=\frac{X_L}{R}=\frac{\pi}{1}=\pi\ \Rightarrow\ \phi=\tan^{-1}(\pi)\approx72.3^\circ.

Converting to radians for the time calculation,

\phi=72.3^\circ\times\frac{\pi}{180^\circ}\approx1.263\text{ rad}.

4. Time lag

The angular frequency of the supply is …

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Q.A coil of $0.01\ \text{H}$ inductance and $1\ \Omega$ resistance is connected to $200\ \text{V}$ , $50\ \text{Hz}$ ac supply. Find the impedance of the circuit and time lag between maximum alternating voltage and current.

Concept understanding — Inductive Reactance Change

Inductive Reactance Change – A First Look

The Core Intuition

The Precise Statement

What "Inductive Reactance Change" Means

A Simple Example …

Why this formula?

Inductive Reactance Change: Why the Formula Holds

1. The Core Idea: Opposition to Current Change

2. Applying a Sinusoidal Current

3. The Phase Shift: Voltage Leads Current

4. Defining Inductive Reactance

5. Why It Changes with Frequency

1. Inductive reactance

2. Impedance

3. Phase angle

4. Time lag

Method: Finding the Impedance and Time Lag Between Voltage and Current in a Series RL (or RC) AC Circuit

Steps

Q.A coil of 0.01 H0.01\ \text{H}0.01 H inductance and 1 Ω1\ \Omega1 Ω resistance is connected to 200 V200\ \text{V}200 V, 50 Hz50\ \text{Hz}50 Hz ac supply. Find the impedance of the circuit and time lag between maximum alternating voltage and current.

1. Inductive reactance

2. Impedance

3. Phase angle

4. Time lag

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Q.A coil of $0.01\ \text{H}$ inductance and $1\ \Omega$ resistance is connected to $200\ \text{V}$ , $50\ \text{Hz}$ ac supply. Find the impedance of the circuit and time lag between maximum alternating voltage and current.

Unlock everything free for 14 days