📌 1. Question Description
⚡ Inductance
Inductance is the property of an electrical circuit by virtue of which
it opposes any change in the current flowing through it.
Φ ∝ I
The induced EMF produced due to changing current is related to the
rate of change of current.
ε = −L dI/dt
⭐ The negative sign represents Lenz's law: the induced EMF
opposes the change in current.
🔵 2. Self-Induction
When the current flowing through a coil changes, the magnetic flux
linked with the same coil changes. This produces an induced EMF in
the same coil. This phenomenon is called self-induction.
ε = −L dI/dt
Here L is called the coefficient of self-inductance.
📐 3. Derivation of Self-Inductance of a Long Solenoid
Consider a long solenoid having:
- Number of turns = N
- Length = l
- Area of cross-section = A
- Current = I
Magnetic field inside a long solenoid is:
B = μ₀nI
where:
n = N/l
Therefore:
B = μ₀NI/l
Magnetic flux through each turn:
Φ = BA
Hence total flux linkage:
NΦ = N(BA)
Substituting B:
NΦ = N(μ₀NI/l)A
By definition:
NΦ = LI
Therefore:
L = μ₀N²A/l
✔ Self-inductance of a long air-core solenoid:
L = μ₀N²A/l
🧲 4. Solenoid with Magnetic Core
If a magnetic material having relative permeability μᵣ is inserted
inside the solenoid:
L = μ₀μᵣN²A/l
Therefore increasing the permeability of the core increases the
self-inductance.
🔬 5. Animated Practical — Self-Induction
↑
Changing current → Induced EMF produced
🧪 6. Practical Experiment — Self-Induction
Experiment: Coil and Battery
Apparatus:
- Inductor/coil
- Battery
- Switch
- Galvanometer or LED
- Connecting wires
Procedure:
- Connect the coil with a battery and switch.
- Close the switch and observe the current.
- Open the switch suddenly.
- Observe the induced EMF in the coil.
✔ Changing current produces changing magnetic flux.
✔ Changing flux induces EMF in the same coil.
✔ The induced EMF opposes the change in current.
🟣 7. Mutual Induction
When a changing current in one coil produces a changing magnetic flux
through a nearby second coil, an induced EMF is produced in the second
coil. This phenomenon is called mutual induction.
ε₂ = −M dI₁/dt
Here M is the coefficient of mutual inductance between the two
coils.
🔬 8. Animated Practical — Mutual Induction
🟡 Primary coil current changes → 🔵 Magnetic flux changes →
🟢 EMF induced in secondary coil
📐 9. Derivation of Mutual Inductance of Two Long Coaxial Solenoids
Consider two long coaxial solenoids wound over the same region.
Let:
- Primary turns = N₁
- Secondary turns = N₂
- Length = l
- Common area = A
- Primary current = I₁
Magnetic field produced by the primary solenoid:
B₁ = μ₀N₁I₁/l
Magnetic flux through each turn of the secondary:
Φ₂ = B₁A
Therefore total flux linkage with secondary:
N₂Φ₂ = N₂B₁A
Substituting B₁:
N₂Φ₂ =
μ₀N₁N₂AI₁/l
By definition:
N₂Φ₂ = MI₁
Hence:
M = μ₀N₁N₂A/l
✔ Mutual inductance of two long coaxial solenoids:
M = μ₀N₁N₂A/l
🔗 10. Relation Between L, M and Coupling Coefficient
If two coils have self-inductances L₁ and L₂ and coefficient of
coupling k, then:
M = k√(L₁L₂)
For perfect coupling:
k = 1
Therefore maximum mutual inductance is:
M = √(L₁L₂)
⚡ 11. Energy Stored in an Inductor
An inductor stores energy in its magnetic field.
U = ½LI²
Energy density in a magnetic field is:
u = B²/(2μ₀)
⚙️ 12. Applications of Inductance
🔌 Transformers
Mutual induction is the basic working principle of transformers.
📻 Radio Tuning
Inductors are used in LC circuits for selecting desired frequencies.
⚡ Choke Coil
Inductors can limit AC current with comparatively small power loss.
🔋 Energy Storage
Inductors store energy in their magnetic fields.
🔄 Oscillators
Inductors are used with capacitors in LC oscillatory circuits.
🛡️ Filters
Inductors are used in electrical and electronic filtering circuits.
📊 13. Self-Induction vs Mutual Induction
| Feature |
Self-Induction |
Mutual Induction |
| Number of coils |
One coil |
Two coils |
| Induced EMF |
In the same coil |
In another coil |
| Formula |
ε = −L dI/dt |
ε₂ = −M dI₁/dt |
| Coefficient |
L |
M |
| Main application |
Choke coils, filters |
Transformers |
📚 14. Important Formulae
ε = −L dI/dt
L = NΦ/I
L = μ₀N²A/l
L = μ₀μᵣN²A/l
ε₂ = −M dI₁/dt
M = N₂Φ₂/I₁
M = μ₀N₁N₂A/l
M = k√(L₁L₂)
U = ½LI²
📝 15. MCQ Practice — 15 Questions
1. The SI unit of inductance is:
A. Tesla
B. Weber
C. Henry
D. Farad
✔ Answer: C
2. Self-induced EMF is given by:
A. ε = LI
B. ε = −L dI/dt
C. ε = IR
D. ε = BLv
✔ Answer: B
3. The self-inductance of a solenoid is proportional to:
A. N
B. N²
C. 1/N
D. √N
✔ Answer: B
4. The self-inductance of a long solenoid is:
A. μ₀NA/l
B. μ₀N²A/l
C. μ₀Nl/A
D. μ₀A/N²l
✔ Answer: B
5. Mutual induction occurs between:
A. Two capacitors
B. Two coils
C. Two resistors
D. Two batteries
✔ Answer: B
6. The SI unit of mutual inductance is:
A. Henry
B. Ohm
C. Volt
D. Tesla
✔ Answer: A
7. Mutual inductance of two coaxial solenoids is proportional to:
A. N₁N₂
B. N₁/N₂
C. N₁+N₂
D. 1/N₁N₂
✔ Answer: A
8. The coefficient of coupling k lies between:
A. 0 and 1
B. 1 and 2
C. −1 and 0
D. 2 and 10
✔ Answer: A
9. For perfect coupling:
A. k = 0
B. k = 0.5
C. k = 1
D. k = 2
✔ Answer: C
10. Energy stored in an inductor is:
A. LI²
B. ½LI²
C. 2LI²
D. L/I²
✔ Answer: B
11. Mutual induction is the basic principle of:
A. Transformer
B. Ammeter
C. Voltmeter
D. Galvanometer only
✔ Answer: A
12. Increasing the number of turns of a solenoid increases its:
A. Resistance only
B. Self-inductance
C. Capacitance only
D. Temperature only
✔ Answer: B
13. The negative sign in ε = −L dI/dt represents:
A. Ohm's law
B. Lenz's law
C. Coulomb's law
D. Ampere's law
✔ Answer: B
14. If current through an inductor is constant, induced EMF is:
A. Maximum
B. Zero
C. Infinite
D. Negative infinity
✔ Answer: B
15. The dimension of inductance is:
A. ML²T⁻²A⁻²
B. ML²T⁻²A⁻¹
C. MLT⁻¹A⁻¹
D. ML⁻¹T⁻²
✔ Answer: A
🟣 16. Assertion–Reason — 5 Questions
Options:
A. Both A and R are true, and R is the correct explanation of A.
B. Both A and R are true, but R is not the correct explanation of A.
C. A is true, but R is false.
D. A is false, but R is true.
Assertion: A coil opposes a change in current through it.
Reason: Self-induced EMF opposes the change in current.
Answer: A
Assertion: Self-inductance of a solenoid is proportional to N².
Reason: Both the magnetic field and flux linkage increase with
the number of turns.
Answer: A
Assertion: A steady current produces no self-induced EMF.
Reason: Self-induced EMF depends on dI/dt.
Answer: A
Assertion: Mutual inductance is measured in henry.
Reason: Mutual inductance is the ratio of flux linkage in one
coil to current in the other coil.
Answer: A
Assertion: Mutual inductance is maximum when coupling between
two coils is perfect.
Reason: For perfect coupling, k = 1.
Answer: A
🟢 17. 2 Marks — 6 Questions
Q1
Define self-induction and write its mathematical expression.
Q2
Define mutual induction and write the expression for induced EMF.
Q3
Define coefficient of self-inductance. Give its SI unit.
Q4
What is the SI unit of mutual inductance?
Q5
What is the physical significance of the negative sign in
ε = −L dI/dt?
Q6
Write the expression for energy stored in an inductor.
🟡 18. 3 Marks — 6 Questions
Q1
Explain the phenomenon of self-induction.
Q2
Explain mutual induction between two coils.
Q3
Write three factors affecting the self-inductance of a solenoid.
Q4
What is coefficient of coupling? State its range.
Q5
Explain why an inductor opposes sudden changes in current.
Q6
Write the expression for energy stored in an inductor and explain
each term.
🟠 19. 4 Marks — 6 Questions
Q1
Derive the expression for self-inductance of a long solenoid.
Q2
Derive the expression for mutual inductance of two long coaxial
solenoids.
Q3
Explain the factors affecting the self-inductance of a solenoid.
Q4
Explain the relation M = k√(L₁L₂).
Q5
Explain the working principle of a transformer using mutual induction.
Q6
Explain the energy stored in an inductor.
🔴 20. 5 Marks — 6 Questions
Q1
Derive the expression L = μ₀N²A/l for a long solenoid.
Q2
Derive the expression M = μ₀N₁N₂A/l for two long coaxial solenoids.
Q3
Explain self-induction, induced EMF and the significance of
self-inductance.
Q4
Explain mutual induction and discuss its applications.
Q5
Derive the expression for energy stored in an inductor.
Q6
Explain coefficient of coupling and derive M = k√(L₁L₂).
🔵 21. 6 Marks — 6 Questions
Q1
Derive the coefficient of self-inductance of a long solenoid
step-by-step and discuss the factors affecting it.
Q2
Derive the coefficient of mutual inductance of two long coaxial
solenoids and explain the physical meaning of mutual inductance.
Q3
Explain self-induction using Lenz's law and derive
ε = −L dI/dt.
Q4
Explain mutual induction, coefficient of coupling and the relation
M = k√(L₁L₂).
Q5
Derive the expression for energy stored in an inductor and obtain
the magnetic energy density.
Q6
Compare self-induction and mutual induction with their mathematical
expressions, solenoid derivations and practical applications.
🧮 22. Numerical Practice — 5 Questions
Q1. A solenoid has 1000 turns, length 0.5 m and cross-sectional
area 4 × 10⁻⁴ m². Calculate its self-inductance.
Q2. The current through a 2 H inductor changes at a rate of
3 A/s. Find the magnitude of induced EMF.
Q3. A 4 H inductor carries a current of 2 A. Calculate the
energy stored in it.
Q4. Two coils have self-inductances 4 H and 9 H. If their
coefficient of coupling is 0.5, calculate their mutual inductance.
Q5. Two coaxial solenoids have 500 and 1000 turns respectively,
common area 2 × 10⁻⁴ m² and length 0.4 m. Calculate their mutual
inductance assuming an air core.
🧪 23. Practical Applications
🔌 Transformer
Works on mutual induction between primary and secondary coils.
⚡ Choke Coil
Uses inductive reactance to control alternating current.
📻 Tuning Circuit
Inductor and capacitor form an LC circuit used for frequency
selection.
🔋 Energy Storage
An inductor stores energy in its magnetic field.
🚀 24. Quick Revision
Self-Induction
Induced EMF is produced in the same coil.
ε = −L dI/dt
Mutual Induction
EMF is induced in a neighbouring coil.
ε₂ = −M dI₁/dt
Two Solenoids
M = μ₀N₁N₂A/l
