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Electricity and Magnetism
Self-Inductance
Self-inductance is the property of a single electrical conductor or coil by which a change in electric current flowing through it induces an electromotive force (EMF) in the same conductor. This phenomenon is a direct consequence of Faraday's law of electromagnetic induction and Lenz's law.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of self-inductance, magnetic flux linkage, formula L = Nphi/I, self-inductance of a long straight solenoid, simple numericals on inductance, and qualitative understanding of back-EMF.
PGT extension
Detailed field-theoretic derivations of self-inductance for coaxial cables and toroids, stored magnetic energy density, transient growth and decay of current in LR circuits, and coupling with AC impedance phenomena.
Prerequisites
Faraday's laws of electromagnetic induction
Lenz's law and direction of induced EMF
Ampere's circuital law and magnetic field of solenoids
Origin and Definition of Self-Inductance
When an electric current I passes through a coil of N turns, it produces a magnetic flux phi through each turn.
The total magnetic flux linkage (Nphi) is directly proportional to the current I flowing through the coil, provided the magnetic permeability of the medium remains constant.
Mathematically expressed as Nphi = L I, where L is the coefficient of self-induction or simply self-inductance.
Back-EMF and Lenz's Law Interpretation
Any change in current (dI/dt) causes a change in magnetic flux linkage, inducing an electromotive force e given by $e = - d(Nphi)/dt = - L (dI/dt)$.
The negative sign embodies Lenz's law, signifying that the induced EMF opposes the change in current that produces it.
For a growing current, the induced EMF acts in the opposite direction to the applied voltage, while for a decaying current, it acts in the same direction to maintain the current.
Dimensions and SI Units
The SI unit of self-inductance is the henry (H), equivalent to volt-second per ampere (V·s/A) or ohm-second.
The dimensional formula of self-inductance is [M L^2 T^{-2} A^{-2}].
An inductor of 1 henry induces a back-EMF of 1 volt when the current changes at the rate of 1 ampere per second.
Magnetic Energy Stored in an Inductor
Work must be done by an external source to establish current against the opposing back-EMF in an inductor.
This instantaneous work is stored within the magnetic field of the inductor as potential energy.
The total energy stored when a current I is established is given by $U = (1/2) L I^2$.
Derivations
Self-Inductance of a Long Solenoid
Conditions: Long air-cored solenoid of length l, cross-sectional area A, total turns N, and uniform turn density $n = N/l$.
Write the magnetic field inside an ideal long solenoid carrying current I: $B = \mu_0 n I = \mu_0 (N/l) I$.
Calculate the magnetic flux through a single turn of the solenoid: ph$i = B A = (\mu_0 N I A) / l$.
Multiply by the total number of turns N to get the total flux linkage: Nph$i = N [(\mu_0 N I A) / l] = (\mu_0 N^2 A / l) I$.
Compare with the defining relation Nphi = L I to extract the self-inductance expression.
Conditions: Inductor of self-inductance L carrying a time-varying current i(t).
Express the instantaneous power delivered to the inductor to overcome back-EMF: $P = -e \cdot i = (L \frac{di}{dt}) i$.
Express the small work done dW in time dt as dW = P dt = L i di.
Integrate the work from zero current to final current I to find total stored energy: $U = \int_{0}^{I} L i di$.
Evaluate the integral to yield the final energy formula.
Result: $U = \frac{1}{2} L I^2$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Magnetic Flux Linkage Definition
$$Nphi = L I$$
N = number of turns, phi = flux per turn, $L = self-inductance$, I = current Weber (Wb)
Linear magnetic medium
Induced EMF in an Inductor
$$e = -L \frac{di}{dt}$$
e = induced EMF, $L = self-inductance$, di/dt = rate of change of current Volt (V)
L is constant
Self-Inductance of a Solenoid
$$L = \frac{\mu_0 N^2 A}{l}$$
m$u_0 = permeability of free space$, N = turns, A = area, l = length Henry (H)
Air-cored, length much greater than radius (l >> r)
Self-Inductance of a Toroid
$$L = \frac{\mu_0 N^2 A}{2 \pi r_{avg}}$$
N = total turns, $A = cross-sectional area$, $r_{avg} = mean radius$ Henry (H)
Close-wound toroidal coil
Stored Magnetic Potential Energy
$$U = \frac{1}{2} L I^2$$
U = energy, $L = self-inductance$, I = steady current Joule (J)
Constant inductance
Magnetic Energy Density
$$u_B = \frac{B^2}{2 \mu_0}$$
$u_B = energy per unit volume$, B = magnetic field Joule per cubic meter (J/m^3)
Uniform magnetic field in free space
Growth of Current in LR Circuit
$$i(t) = \frac{V}{R} (1 - e^{-Rt/L})$$
V = applied DC voltage, R = resistance, L = inductance, t = time Ampere (A)
Circuit closed at t = 0
Time Constant of LR Circuit
$$\tau = \frac{L}{R}$$
\tau = time constant, $L = self-inductance$, R = resistance Second (s)
Series LR circuit
Graphs and experiments
Current Growth in an LR Circuit
Draw: An exponential growth curve of current i versus time t, asymptotically approaching the maximum value $I_0 = V/R$, with the tangent at $t=0 intercepting the asymptote at t = L/R$.
Interpret: Shows how inertia of the inductor delays the instantaneous rise of current due to opposing back-EMF.
Exam use: Directly tested in conceptual MCQs regarding time constant and exponential rise.
Magnetic Flux vs Current Graph
Draw: A straight line passing through the origin for linear magnetic materials, and a hysteresis/saturation curve for ferromagnetic cores.
Interpret: The slope of the linear graph directly represents the constant self-inductance L of the inductor.
Exam use: Used to test understanding of linearity and core saturation effects in inductors.
Induced EMF vs Rate of Change of Current
Draw: A linear plot of magnitude of induced EMF (|e|) versus di/dt with a positive slope equal to L.
Interpret: Demonstrates strict proportionality between the back-EMF and the rate of change of current.
Exam use: Numerical data interpretation in TGT/PGT problem sets.
Solved numericals
TGT core
A coil has 500 turns and a current of 2 A produces a magnetic flux of 4 x 10^{-3} Wb through each turn. Calculate the self-inductance of the coil.
Identify given values: N = 500, I = 2 A, ph$i = 4 \times 10^{-3} Wb$.
Apply the formula for magnetic flux linkage: Nph$i = 500 \times 4 \times 10^{-3} = 2.0 Wb$.
Use the definition $L = Nphi / I$.
Calculate $L = 2.0 / 2 = 1.0 H$.
Answer: 1.0 H
TGT core
An inductor of inductance 5 H carries a steady current of 2 A. What is the magnetic energy stored in the inductor?
Identify given parameters: L = 5 H, I = 2 A.
Recall the formula for stored magnetic energy: $U = (1/2) L I^2$.
A solenoid of length 0.5 m, cross-sectional area 10 cm^2, and 1000 turns is filled with a magnetic material of relative permeability m$u_r = 500$. Calculate its self-inductance.
Convert dimensions to SI units: l = 0.5 m, $A = 10 \times 10^{-4} m^2 = 10^{-3} m^2$, N = 1000, m$u_r = 500$.
Use the formula for self-inductance with a core medium: $L = (\mu_r \mu_0 N^2 A) / l$.
Calculate final value: $L = 1000 \times 4pi \times 10^{-4} = 0.4pi H \approx 1.256 H$.
Answer: 1.256 H (or 0.4pi H)
Assumptions and limitations
Self-inductance L is assumed to be constant, which holds true only for linear magnetic materials (mu is independent of B).
Fringing fields at the ends of finite solenoids are neglected in standard analytical formulas, requiring long solenoid approximations (l >> r).
Common misconceptions
Common error: An inductor opposes the flow of steady direct current (DC).
Correct: An ideal inductor offers zero resistance and acts as a short circuit for steady DC in steady state.
Inductance opposes only the change in current (di/dt), not the steady current itself.
Common error: Self-inductance depends on the magnitude of current passing through the coil.
Correct: Self-inductance is a purely geometric property of the conductor configuration and core material permeability.
In linear media, flux linkage Nphi increases proportionally with I, keeping the ratio $L = Nphi / I constant$.
Quick revision
Self-inductance L is defined by Nphi = L I and measured in henries (H).
Magnetic Flux Linkage Definition: Nphi = L I
Induced EMF in an Inductor: $e = -L \frac{di}{dt}$
Must remember
Total magnetic flux linkage is directly proportional to current: $N\phi = L I$.
Induced back-EMF is given by $e = -L \frac{di}{dt}$, embodying Lenz's law.
SI unit of self-inductance is the henry ($H$), equivalent to $V\cdot s/A$ or $\Omega\cdot s$.
Dimensional formula of self-inductance is $[M L^2 T^{-2} A^{-2}]$.}
Magnetic energy stored in an inductor is $U = \frac{1}{2} L I^2$.
Self-inductance of a long air-cored solenoid is $L = \frac{\mu_0 N^2 A}{l}$.
Time constant of a series LR circuit is $\tau = \frac{L}{R}$.
Self-inductance is a geometric property and does not depend on the magnitude of the current.
Glossary
Self-Inductance ($L$)The property of a coil by virtue of which it opposes any change in the strength of electric current flowing through it.
Back-EMF ($e$)The electromotive force induced in a circuit that opposes the primary change in current producing it.
Henry ($H$)The SI unit of inductance; inductance of a circuit in which a current change of one ampere per second induces an EMF of one volt.
Time Constant ($\tau$)The time required for the current in an LR circuit to reach about $63.2\%$ of its maximum steady value.
Inductor in DC vs AC Steady State
Parameter
Steady DC State
High Frequency AC State
Reactance ($X_L$)
Zero ($X_L = 0$)
Very High ($X_L = \omega L$)
Behavior
Acts as a short circuit (ideal conductor)
Acts as an open circuit (blocks high frequencies)
Back-EMF
Zero (since $\frac{di}{dt} = 0$)
Maximum magnitude varying harmonically
Practice Quiz
Choose an option to reveal the explanation.
1. What is the dimensional formula for self-inductance $L$?
Explanation: Since $e = -L \frac{di}{dt}$, $L = \frac{e \cdot dt}{di}$. Dimensional formula of EMF $e$ is $[M L^2 T^{-3} A^{-1}]$, time is $[T]$, and current is $[A]$. Thus, $[L] = \frac{[M L^2 T^{-3} A^{-1}][T]}{[A]} = [M L^2 T^{-2} A^{-2}]$.
2. Which of the following units is equivalent to one henry?
Explanation: From $e = L \frac{di}{dt}$, $L = e \frac{dt}{di}$. The unit is volt $\times$ second / ampere ($V\cdot s/A$).
3. What does the negative sign in the induced EMF equation $e = -L \frac{di}{dt}$ signify?
Explanation: The negative sign embodies Lenz's law, representing the fact that the induced EMF always opposes the change in current that produces it.
4. How does the self-inductance $L$ of a long air-cored solenoid depend on the magnitude of current $I$ passing through it?
Explanation: Self-inductance is a geometric property of the coil and core medium. In linear media, flux linkage $N\phi$ increases proportionally with $I$, making $L = \frac{N\phi}{I}$ constant and independent of $I$.
5. If the number of turns in a solenoid is doubled while keeping its length and cross-sectional area constant, its self-inductance will:
Explanation: For a solenoid, $L = \frac{\mu_0 N^2 A}{l}$. Since $L \propto N^2$, doubling $N$ increases $L$ by a factor of $2^2 = 4$.
6. What is the magnetic potential energy stored in an inductor of inductance $2\text{ H}$ carrying a steady current of $4\text{ A}$?
Explanation: Using the formula $U = \frac{1}{2} L I^2$, we get $U = \frac{1}{2} (2\text{ H}) (4\text{ A})^2 = 1 \times 16 = 16\text{ J}$.
7. What is the behavior of an ideal inductor in a DC circuit after a long time (steady state)?
Explanation: In steady-state DC, $\frac{di}{dt} = 0$, so the induced back-EMF $e = 0$. An ideal inductor acts as a plain conductor (short circuit) with zero resistance.
8. The time constant of an LR circuit represents the time in which the growing current reaches what fraction of its maximum steady value?
Explanation: For growth of current $i(t) = \frac{V}{R}(1 - e^{-Rt/L})$. When $t = \tau = \frac{L}{R}$, $i(\tau) = \frac{V}{R}(1 - e^{-1}) \approx 0.632 \frac{V}{R}$.
9. What is the expression for the magnetic energy density $u_B$ in free space?
Explanation: Magnetic energy density is defined as energy per unit volume, given by $u_B = \frac{B^2}{2\mu_0}$.
10. If the core of an inductor is filled with a ferromagnetic material of relative permeability $\mu_r$, its self-inductance will:
Explanation: Since $L = \frac{\mu N^2 A}{l}$ and $\mu = \mu_0 \mu_r$, introducing a magnetic core multiplies the inductance by the relative permeability $\mu_r$.
11. An inductor experiences a current changing at the rate of $5\text{ A/s}$. If an induced EMF of $20\text{ V}$ appears across its terminals, what is the self-inductance of the coil?
Explanation: Using magnitude relation $e = L \frac{di}{dt}$, $20 = L \times 5$, which gives $L = \frac{20}{5} = 4\text{ H}$.
12. Which factor does NOT affect the self-inductance of a toroid?
Explanation: Self-inductance of a toroid is $L = \frac{\mu_0 N^2 A}{2\pi r_{\text{avg}}}$, which depends on geometry and core permeability, not on the current magnitude.
13. What happens to the magnetic energy stored in an inductor if the current through it is doubled?
Explanation: Since energy $U = \frac{1}{2} L I^2$, energy is proportional to $I^2$. Doubling $I$ multiplies energy by $2^2 = 4$.
14. In a series LR circuit connected to a DC source, what is the rate of growth of current at the exact instant the circuit is closed ($t = 0$)?
Explanation: At $t = 0$, the current $i = 0$, so the voltage drop across resistor is zero. The entire applied voltage appears across the inductor: $V = L \left(\frac{di}{dt}\right)_{t=0}$, yielding $\left(\frac{di}{dt}\right)_{t=0} = \frac{V}{L}$.
15. Which of the following correctly defines self-induction?
Explanation: Self-induction is the production of an opposing induced EMF in a coil resulting from a change in electric current flowing through that same coil.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The self-inductance $L$ of a solenoid of length $l$, area $A$, and total turns $N$ is proportional to:
Explanation: From the formula $L = \frac{\mu_0 N^2 A}{l}$, self-inductance is directly proportional to the square of the number of turns ($N^2$).
2. Two solenoids have the same length and radius, but solenoid A has twice the number of turns as solenoid B. The ratio of their self-inductances $L_A / L_B$ is:
Explanation: Since $L \propto N^2$ for identical dimensions, $\frac{L_A}{L_B} = \left(\frac{N_A}{N_B}\right)^2 = (2)^2 = 4:1$.
3. A coil has a self-inductance of $10\text{ H}$ and a resistance of $2\Omega$. What is the time constant of the circuit?
5. The current in a coil changes from $2\text{ A}$ to $12\text{ A}$ in $0.2\text{ s}$, resulting in an induced EMF of $50\text{ V}$. What is the self-inductance of the coil?
Explanation: Change in current $\Delta I = 12 - 2 = 10\text{ A}$, time $\Delta t = 0.2\text{ s}$. Rate $\frac{di}{dt} = \frac{10}{0.2} = 50\text{ A/s}$. Using $e = L \frac{di}{dt} \implies 50 = L \times 50 \implies L = 1\text{ H}$.
6. What is the unit of the ratio $\frac{L}{R}$ in an LR circuit?
Explanation: The ratio $\frac{L}{R}$ represents the time constant $\tau$ of an LR circuit, whose SI unit is the second.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the current passing through a choke coil of self-inductance $5\text{ H}$ is decreased at a uniform rate of $2\text{ A/s}$, the induced EMF is:
Explanation: Given $L = 5\text{ H}$ and $\frac{di}{dt} = -2\text{ A/s}$ (decreasing current). Using $e = -L \frac{di}{dt} = -(5)(-2) = +10\text{ V}$. For decaying current, induced EMF acts in the same direction to oppose the decrease.
2. A toroid has a mean radius $r_{\text{avg}} = 0.1\text{ m}$, cross-sectional area $A = 10^{-4}\text{ m}^2$, and total turns $N = 1000$. Its self-inductance is approximately:
3. Which of the following statements regarding self-inductance is correct?
Explanation: Self-inductance is analogous to mechanical inertia (mass); it opposes any change in current (rate of change of current), storing energy in a magnetic field.
4. A coil of resistance $10\Omega$ and inductance $5\text{ H}$ is connected to a $100\text{ V}$ battery. What is the final steady current in the circuit?
Explanation: In steady state, the inductor acts as a short circuit ($e = 0$). The current is determined purely by Ohm's law: $I_{\max} = \frac{V}{R} = \frac{100}{10} = 10\text{ A}$.
5. What is the magnetic energy density inside a solenoid where the magnetic field is $B = 0.2\text{ T}$?
Explanation: Energy density is $u_B = \frac{B^2}{2\mu_0} = \frac{(0.2)^2}{2\mu_0} = \frac{0.04}{2\mu_0}$.
6. If a current of $2\text{ A}$ produces a magnetic flux linkage of $4\text{ Wb}$ in a coil, its self-inductance is:
Explanation: Using $N\phi = L I$, we find $L = \frac{N\phi}{I} = \frac{4}{2} = 2\text{ H}$.
7. In an LR circuit, the time constant $\tau$ depends on:
Explanation: The time constant is given by $\tau = \frac{L}{R}$, which depends directly on both $L$ and $R$.
8. Why does a spark sometimes occur across the switch contacts when a circuit containing a large inductor is suddenly opened?
Explanation: When opening a switch rapidly, $\frac{di}{dt}$ is extremely large, creating a massive induced EMF ($e = -L \frac{di}{dt}$) that can break down air insulation, causing a spark.
9. Two inductors of self-inductance $L_1$ and $L_2$ are connected in series opposition such that their mutual inductance can be neglected. Their equivalent inductance is:
Explanation: When mutual inductance is neglected (or independent coils in series), inductances simply add up: $L_{\text{eq}} = L_1 + L_2$.
10. The self-inductance of a coil depends on which of the following?
Explanation: Self-inductance is solely a geometric property determined by the physical dimensions of the conductor configuration and the magnetic permeability of the surrounding or core medium.