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Electricity and Magnetism
Mutual Inductance
Mutual inductance is the phenomenon where a change in current in one coil induces an electromotive force in a nearby secondary coil due to changing magnetic flux linking them. Its SI unit is the Henry (H).
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of mutual inductance, factors affecting mutual inductance, formula for mutual inductance of two long coaxial solenoids, and numerical problems based on M = E / (dI/dt) and M = N2Φ2 / I1.
PGT extension
Neumann's formula for mutual inductance, reciprocity theorem (M12 = M21), coefficient of coupling (k), series and parallel combinations of mutually coupled inductors, and energy stored in a system of two coupled coils.
Prerequisites
Faraday's laws of electromagnetic induction
Magnetic flux and Ampere's circuital law
Self-inductance of a coil
Definition and Fundamental Principle
When an electric current I1 flows through a primary coil, it produces a magnetic field that links with a secondary coil, generating a magnetic flux Φ2 through each turn of the secondary.
The total magnetic flux linkage N2Φ2 in the secondary coil is directly proportional to the current I1 in the primary coil, expressed as N2Φ2 = M12 I1.
The proportionality constant M12 is defined as the mutual inductance of coil 2 with respect to coil 1.
Neumann's Reciprocity Theorem
The mutual inductance between two magnetically coupled circuits is symmetric, meaning M12 = M21 = M.
This symmetry holds true in the absence of magnetic saturation and ferromagnetic hysteresis, depending entirely on the geometrical configuration and medium permeability.
It implies that the flux linking coil 2 per unit current in coil 1 is identical to the flux linking coil 1 per unit current in coil 2.
Coefficient of Coupling
The coefficient of coupling (k) measures the fraction of magnetic flux produced by the primary coil that successfully links with the secondary coil.
It is defined mathematically as $k = M / \sqrt(L1 L2)$, where L1 and L2 are the self-inductances of the two coils.
For tightly coupled coils (like in an ideal transformer with an iron core), k = 1; for loosely coupled coils separated in air, k is significantly less than 1.
Combination of Mutually Coupled Inductors
When two coils of self-inductances L1 and L2 and mutual inductance M are connected in series aiding (fluxes in same direction), the equivalent inductance is Le$q = L1 + L2 + 2M$.
When connected in series opposing (fluxes in opposite directions), the equivalent inductance is Le$q = L1 + L2 - 2M$.
This difference provides a standard experimental method to determine the value of mutual inductance M.
Derivations
Mutual Inductance of Two Long Coaxial Solenoids
Conditions: Two long coaxial solenoids of same length l, radii r1 and r2 (r1 < r2), and turn densities n1 and n2 per unit length.
Let a current I1 flow through the inner solenoid (solenoid 1). The magnetic field B1 produced inside solenoid 1 is given by B$1 = \mu_0 n1 I1$.
Since the outer solenoid (solenoid 2) encloses solenoid 1, the magnetic flux Φ2 through each turn of solenoid 2 is equal to B1 multiplied by the cross-sectional area of the inner solenoid, Φ$2 = B1 * (\pi r1^2) = \mu_0 n1 I1 \pi r1^2$.
The total magnetic flux linkage of the outer solenoid having N2 = n2 l turns is N2 Φ$2 = (n2 l) (\mu_0 n1 I1 \pi r1^2)$.
Using the definition N2 Φ2 = M I1, equate the expressions to find M.
Result: $M = \mu_0 n1 n2 \pi r1^2 l$
Equivalent Inductance of Two Coils in Series Aiding and Opposing
Conditions: Two coils with self-inductances L1, L2 and mutual inductance M connected in series.
Let current i flow through the series combination. The voltage drop across coil 1 is V$1 = L1(di/dt) + M(di/dt)$ and across coil 2 is V$2 = L2(di/dt) + M(di/dt)$ for series aiding.
The total voltage V across the combination is $V = V1 + V2 = (L1 + L2 + 2M)(di/dt)$.
Comparing with $V = Leq(di/dt) gives the series aiding formula$.
For series opposing, the induced EMF opposes the self-induced EMF, resulting in $V = (L1 + L2 - 2M)(di/dt)$.
U = total stored magnetic energy, I1, I2 = currents, M = mutual inductance Joule (J)
Linear coupled system
Graphs and experiments
Induced EMF vs Rate of Change of Current
Draw: A straight line passing through the origin in an |e2| versus (dI1/dt) plot.
Interpret: The slope of the straight line yields the magnitude of the mutual inductance M.
Exam use: Tested in conceptual interpretation and gradient-based numerical calculation.
Series Aiding and Opposing Inductance Bridge
Draw: Experimental setup schematic showing two coupled coils connected in series with a source and AC bridge.
Interpret: Measuring Leq in both polarities allows direct extraction of M via (L_aiding - L_opposing) / 4.
Exam use: Frequently asked in experimental physics viva and conceptual PGT MCQs.
Coefficient of Coupling Flux Leakage Graph
Draw: Plot of k versus physical separation distance d between two identical coils.
Interpret: k drops sharply with distance as leakage flux increases and magnetic coupling weakens.
Exam use: Conceptual questions on transformer core design and air-core inductors.
Solved numericals
TGT core
The mutual inductance of two coils is 1.5 H. If the current in the primary coil changes from 10 A to 2 A in 0.05 seconds, calculate the induced EMF in the secondary coil.
Identify given values: M = 1.5 H, initial current I1_i = 10 A, final current I1_f = 2 A, dt = 0.05 s.
Calculate the change in current: dI$1 = I1_f - I1_i = 2 - 10 = -8 A$.
Apply the formula e$2 = -M (dI1 / dt): e2 = -1.5 * (-8 / 0.05)$.
Two coils connected in series show an equivalent inductance of 3.8 H in series aiding and 1.6 H in series opposing. Calculate the self-inductance of each coil assuming they are identical (L1 = L2 = L) and find their mutual inductance.
Write down the series aiding and opposing equations: Leq$1 = 2L + 2M = 3.8$ and Leq$2 = 2L - 2M = 1.6$.
Subtract the second equation from the first: 4$M = 3.8 - 1.6 = 2.2$, which gives $M = 2.2 / 4 = 0.55 H$.
Add the two equations: 4$L = 3.8 + 1.6 = 5.4$, which gives $L = 5.4 / 4 = 1.35 H$.
Verify: L1 = 1.35 H, L2 = 1.35 H, M = 0.55 H.
Answer: L1 = 1.35 H, L2 = 1.35 H, M = 0.55 H
PGT extension
Two coaxial solenoids each of length 0.5 m have radii 2 cm and 3 cm respectively. The inner solenoid has 500 turns and the outer solenoid has 1000 turns. Find the mutual inductance between them. (Take m$u_0 = 4*\pi * 10^-7 T m/A)$
Mutual inductance is assumed constant and independent of the magnitude of current only in linear magnetic media without ferromagnetic saturation.
Neumann's reciprocity formula assumes absence of displacement currents and non-magnetic surrounding media (m$u_r = 1)$.
Common misconceptions
Common error: Mutual inductance depends on the rate of change of current (dI/dt).
Correct: Mutual inductance is a geometrical and material property independent of current or its rate of change.
Induced EMF depends on dI/dt, but the mutual inductance M is strictly a function of coil geometry, relative position, and core permeability.
Common error: Coefficient of coupling k can exceed 1 if the secondary has more turns than the primary.
Correct: The coefficient of coupling k can never exceed 1.
k represents the ratio of linked flux to total flux, bounded strictly between 0 and 1 regardless of turn ratios.
Quick revision
$M = N2Φ2 / I1 = -e2 / (dI1/dt) = M12 = M21$
Series aiding: Le$q = L1 + L2 + 2M; Series opposing: Leq = L1 + L2 - 2M$
Coefficient of coupling: $k = M / \sqrt(L1 L2) <= 1$
Must remember
Mutual inductance $M$ is strictly a geometrical, positional, and material property independent of current or rate of change of current.
Neumann's Reciprocity Theorem states that $M_{12} = M_{21} = M$ in linear magnetic media.
The coefficient of coupling is defined as $k = \frac{M}{\sqrt{L_1 L_2}}$ and is bounded as $0 \le k \le 1$.
For series aiding connection, equivalent inductance is $L_{eq} = L_1 + L_2 + 2M$; for series opposing, $L_{eq} = L_1 + L_2 - 2M$.
The SI unit of mutual inductance is the Henry (H) or $\text{Wb/A}$.
Glossary
Mutual Inductance ($M$)The property by which a changing current in one coil induces an electromotive force in an adjacent coil.
Coefficient of Coupling ($k$)The fraction of total magnetic flux produced by one coil that links with the second coil.
Neumann's Reciprocity TheoremThe theorem establishing the symmetry of mutual inductance between two magnetically coupled circuits ($M_{12} = M_{21}$).
Series AidingA connection of two coupled inductors where their magnetic fluxes reinforce each other, increasing total inductance.
Series Aiding vs Series Opposing Inductors
Feature
Series Aiding
Series Opposing
Magnetic Flux Direction
Same direction (reinforcing)
Opposite direction (cancelling)
Equivalent Inductance
$L_{eq} = L_1 + L_2 + 2M$
$L_{eq} = L_1 + L_2 - 2M$
Total Stored Energy
Higher due to positive mutual term
Lower due to negative mutual term
Practice Quiz
Choose an option to reveal the explanation.
1. What is the SI unit of mutual inductance?
Explanation: The SI unit of both self-inductance and mutual inductance is the Henry (H), equivalent to Volt-second per Ampere.
2. The mutual inductance $M$ between two coils depends upon:
Explanation: Mutual inductance is strictly a geometrical and material property determined by the shape, size, distance between coils, and core permeability.
3. According to Neumann's Reciprocity Theorem, for two coupled circuits, the mutual inductances $M_{12}$ and $M_{21}$ are related as:
Explanation: Neumann's Reciprocity Theorem proves that the mutual inductance between two circuits is symmetric, meaning $M_{12} = M_{21}$ in the absence of ferromagnetic saturation.
4. What is the range of values for the coefficient of coupling $k$?
Explanation: The coefficient of coupling $k = \frac{M}{\sqrt{L_1 L_2}}$ represents a fraction of magnetic flux linked, constrained strictly between $0$ and $1$.
5. Two coils are connected in series aiding. If their self-inductances are $L_1$ and $L_2$ and mutual inductance is $M$, the equivalent inductance is:
Explanation: When connected in series aiding, the fluxes reinforce each other, resulting in $L_{eq} = L_1 + L_2 + 2M$.
6. When two coils are connected in series opposing, the equivalent inductance is given by:
Explanation: In series opposing, magnetic fields oppose each other, leading to an equivalent inductance of $L_{eq} = L_1 + L_2 - 2M$.
7. The induced EMF in the secondary coil is expressed as $e_2 = -M \frac{dI_1}{dt}$. What does the negative sign signify?
Explanation: The negative sign is a direct consequence of Lenz's Law, indicating that the induced EMF opposes the change in primary current that produces it.
8. If the primary current changes at a rate of $5\text{ A/s}$ and induces an EMF of $10\text{ V}$ in the secondary, what is the mutual inductance?
Explanation: Using $e_2 = M \frac{dI_1}{dt}$, $10 = M \times 5$, which gives $M = \frac{10}{5} = 2\text{ H}$.
9. For an ideal transformer with an iron core where all flux links both coils, the coefficient of coupling $k$ is:
Explanation: In an ideal transformer with perfect magnetic coupling, all flux links primary and secondary, so $k = 1$.
10. The formula $M = \frac{N_2 \Phi_2}{I_1}$ defines mutual inductance in terms of:
Explanation: Mutual inductance is defined as the total magnetic flux linkage in the secondary coil ($N_2 \Phi_2$) per unit current flowing in the primary ($I_1$).
11. What happens to the mutual inductance between two coils if the distance between them is increased?
Explanation: As distance increases, magnetic flux leakage increases and fewer field lines link the secondary coil, causing $M$ to decrease.
12. Neumann's formula for mutual inductance involves a double line integral over:
Explanation: Neumann's formula $M = \frac{\mu_0}{4\pi} \oint \oint \frac{dl_1 \cdot dl_2}{r}$ is a double line integral over two closed circuits.
13. Which of the following represents the correct expression for the total magnetic energy stored in two magnetically coupled coils?
Explanation: The stored magnetic energy in a linear coupled system accounts for self-energies and mutual interaction energy: $U = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 \pm M I_1 I_2$.
14. If two identical coils each of self-inductance $L$ are connected in series aiding with complete coupling ($k=1$), the equivalent inductance is:
Explanation: With $L_1 = L_2 = L$ and $M = k\sqrt{L_1 L_2} = L$, series aiding gives $L_{eq} = L + L + 2L = 4L$.
15. A common misconception is that mutual inductance depends on the rate of change of current ($dI/dt$). Why is this false?
Explanation: While the magnitude of the induced EMF depends on $dI/dt$, the mutual inductance parameter $M$ itself is independent of current variations.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Two coaxial long solenoids have mutual inductance $M$. If the radius of both solenoids is doubled while keeping their lengths and number of turns per unit length constant, the new mutual inductance will be:
Explanation: For coaxial solenoids, $M = \mu_0 n_1 n_2 \pi r_1^2 l$. Since $r_1$ is doubled, $r_1^2$ becomes $(2r_1)^2 = 4r_1^2$, making the new mutual inductance $4M$.
2. The self-inductances of two coils are $4\text{ H}$ and $9\text{ H}$. If the coefficient of coupling is $0.5$, the mutual inductance between them is:
Explanation: Using $M = k \sqrt{L_1 L_2}$, we get $M = 0.5 \times \sqrt{4 \times 9} = 0.5 \times 6 = 3\text{ H}$.
3. The equivalent inductance of two mutually coupled coils in series aiding is $8\text{ H}$ and in series opposing is $4\text{ H}$. The mutual inductance $M$ between the coils is:
Explanation: Let $L_{eq1} = L_1 + L_2 + 2M = 8$ and $L_{eq2} = L_1 + L_2 - 2M = 4$. Subtracting the second from the first gives $4M = 4$, so $M = 1\text{ H}$.
4. A primary coil has $1000$ turns, and a secondary coil has $500$ turns. A current of $2\text{ A}$ in the primary produces a magnetic flux of $3 \times 10^{-4}\text{ Wb}$ per turn in the secondary. What is the mutual inductance?
5. In a pair of coupled coils, the maximum possible mutual inductance is $6\text{ H}$ and the self-inductances are $9\text{ H}$ and $4\text{ H}$. What is the coefficient of coupling?
Explanation: Using $k = \frac{M}{\sqrt{L_1 L_2}}$, we find $k = \frac{6}{\sqrt{9 \times 4}} = \frac{6}{6} = 1.0$.
6. The current in the primary coil changes from $10\text{ A}$ to $2\text{ A}$ in $0.05\text{ seconds}$. If the mutual inductance is $0.5\text{ H}$, the magnitude of the induced EMF in the secondary coil is:
Explanation: Magnitude of EMF is $|e_2| = M \frac{\Delta I_1}{\Delta t} = 0.5 \times \frac{10 - 2}{0.05} = 0.5 \times \frac{8}{0.05} = 0.5 \times 160 = 80\text{ V}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the permeability of the core material inside two coupled solenoids increases by a factor of 4, the mutual inductance $M$ will:
Explanation: Since $M = \mu_0 \mu_r n_1 n_2 \pi r_1^2 l$, mutual inductance is directly proportional to the relative permeability $\mu_r$ of the core medium.
2. What is the maximum value that the coefficient of coupling $k$ can theoretically attain in physical circuits?
Explanation: The maximum theoretical limit for the coefficient of coupling $k$ is $1.0$, which occurs when all magnetic flux from the primary successfully links the secondary.
3. Two coils with self-inductances $L_1 = 2\text{ H}$ and $L_2 = 8\text{ H}$ have a mutual inductance $M = 3\text{ H}$. What is the coupling coefficient $k$?
4. When two inductors are connected in series opposing, the equivalent inductance is $4\text{ H}$. If connected in series aiding, it is $12\text{ H}$. What are the self-inductances assuming $L_1 = L_2$?
Explanation: $L_1 + L_2 + 2M = 12$ and $L_1 + L_2 - 2M = 4$. Adding gives $2(L_1 + L_2) = 16 \implies L_1 + L_2 = 8$. Since $L_1 = L_2$, each is $4\text{ H}$.
5. Neumann's Reciprocity Theorem fails when:
Explanation: Neumann's theorem $M_{12} = M_{21}$ relies on linear magnetic media and fails in the presence of ferromagnetic saturation or magnetic hysteresis.
6. A primary coil experiences a current change governed by $I_1(t) = 4t^2$. If the mutual inductance is $0.5\text{ H}$, what is the induced EMF in the secondary at $t = 2\text{ s}$?
7. Which factor does NOT affect the mutual inductance between two nearby coils?
Explanation: Mutual inductance depends on magnetic flux linkage and geometry, whereas the resistance of the secondary coil only affects the magnitude of induced current, not the mutual inductance itself.
8. If two identical coils of self-inductance $L$ are connected in series opposing with maximum possible coupling ($k=1$), the equivalent inductance is:
Explanation: With $L_1 = L_2 = L$ and $M = \sqrt{L \cdot L} = L$, series opposing gives $L_{eq} = L + L - 2(L) = 0$.
9. The dimensional formula for mutual inductance is:
Explanation: Since $M = \frac{N \Phi}{I}$ and flux $\Phi = [ML^2 T^{-2} A^{-1}]$, dividing by current $A$ gives $[ML^2 T^{-2} A^{-2}]$, identical to self-inductance.
10. In a coupled coil system, if the primary current is constant ($dI_1/dt = 0$), what is the induced EMF in the secondary coil?
Explanation: According to Faraday's Law and $e_2 = -M \frac{dI_1}{dt}$, if the primary current is constant, the rate of change of current is zero, hence the induced EMF is zero.