Mutual Inductance Formula, Derivation & Real-Life Applications
Mutual inductance is a core concept in electromagnetic induction, explaining how a change in current in one coil causes an emf in another coil placed nearby. Devices such as transformers and wireless chargers rely on this phenomenon for energy transfer without direct connection. In JEE Main Physics, understanding mutual inductance is essential for circuits involving coupled coils, the dot convention, and calculating induced emf.
Whenever the current in one coil changes with time, it varies the magnetic flux through a second coil positioned close to it. According to Faraday’s law of electromagnetic induction, this changing flux induces an emf in the second coil. The measure of how effectively one coil induces emf in another is called mutual inductance. Practical examples include the operation of step-up and step-down transformers, mobile wireless charging, and many experimental physics setups.
Mutual Inductance Law, Formula, and SI Units
The mutual inductance, usually denoted by M, quantifies the emf produced in one coil when the current in the other changes. By Faraday’s law, if two coils are positioned such that the magnetic field of one passes through the other, the law is expressed as:
Induced emf in Coil 2, ε2 = -M (dI1/dt).
Here, M (mutual inductance) depends on coil geometry, number of turns, coil orientation, and medium. Its SI unit is the Henry (H). For JEE Main numericals, always check for unit consistency and correctly apply the sign convention when using the formula.
| Physical Quantity | Symbol | Unit (SI) | Definition |
|---|---|---|---|
| Mutual inductance | M | Henry (H) | EMF induced in one coil due to unit rate change of current in another nearby coil |
Physical Significance and Working Principle of Mutual Inductance
Mutual inductance captures how magnetic flux linkage between coils enables energy transfer. The effect depends on the geometric arrangement, the number of turns in each coil, area of cross-section, and separation distance. The coupling is strongest when coils share a common axis and are wound closely — a crucial idea for transformer operation.
Suppose Coil 1 carries current, creating a magnetic field that passes through Coil 2. When current in Coil 1 changes, the resulting change in flux through Coil 2 induces an emf, expressed using the mutual inductance formula above.
Calculation and Application: Mutual Inductance Between Two Coils
The mutual inductance of two long solenoids, one inside the other, is given by:
M = μ0 n1 n2 A l
where μ0 is magnetic permeability, n1 and n2 are the number of turns per unit length, A is common cross-sectional area, and l is overlap length. For coupled coils like those in transformers, the value of mutual inductance directly influences efficiency.
- Higher number of turns or larger common area increases mutual inductance.
- Greater coil separation or poor alignment reduces mutual inductance.
- Introducing a soft iron core maximizes flux linkage and mutual coupling.
- Used in all transformer-based circuits and wireless energy transfer.
- Directly tested in electromagnetic induction and alternating currents problems.
Example: If a change of 4 A/s in Coil 1 induces 0.80 V in Coil 2, then M = |ε2/(dI1/dt)| = 0.80/4 = 0.20 H.
Mutual Inductance Dot Convention, Circuits, and Comparison with Self-Inductance
In circuit analysis, the dot convention is critical for determining the polarity of induced emf in coupled coils. If current enters the dotted end of one coil, the voltage induced at the dotted end of the second coil is considered positive (otherwise negative).
| Aspect | Mutual Inductance | Self Inductance |
|---|---|---|
| Coils Required | Two separate coils | Single coil |
| Definition | EMF in one coil due to current change in the other | EMF in a coil due to its own current change |
| Dependence on Geometry | Orientation & proximity | Dimensions of coil itself |
| SI Unit | Henry (H) | Henry (H) |
- Use dot convention in Kirchhoff's laws of electric circuits for correct emf sign.
- Questions may combine mutual and self-inductance in transformer numericals.
- Practice with electromagnetic induction practice paper.
Real-Life Examples and Problem-Solving with Mutual Inductance
- Transformer operation in electric power distribution
- Step-up transformer and step-down transformer
- Wireless charging in smartphones and electric vehicles
- Experimental setups for studying alternating currents
- Inductive sensors and coupling devices
- Coaxial solenoids and school lab experiments
Vedantu emphasizes mastery of mutual inductance for JEE Main Physics. For revision, see the electromagnetic induction revision notes and JEE previous year question papers.
- Mutual inductance is affected by the spacing and orientation of coils.
- The sign and value of induced emf depend on the dot convention.
- Soft iron cores increase mutual coupling; air gaps reduce it.
- Practice both theoretical and calculation-based questions.
- Always check units in JEE Main numericals.
Download concise mutual inductance notes and solved problems as a PDF for last-minute JEE revision. Complete mind-maps for electromagnetic induction can be found with Vedantu’s Physics resources.
- Download EM induction revision notes PDF
- Practice EM induction mock test 1
- Important induction questions (PDF)
Mutual inductance bridges basic electromagnetic theory with real circuits. Mastering its laws, formulas, and application cases prepares students for both JEE Main exams and practical engineering scenarios.
FAQs on Mutual Inductance in Physics: Concepts, Laws, and Examples
1. What is mutual inductance?
Mutual inductance is the property of a pair of coils where a change in current in one coil induces an emf in the other nearby coil through electromagnetic induction.
Key points:
- It represents how effectively two coils are magnetically linked.
- Measured in Henry (H).
- Essential for the working of devices like transformers and wireless chargers.
2. Why does mutual inductance happen?
Mutual inductance occurs because a changing current in one coil produces a changing magnetic field, which links to a second coil and induces an emf.
How it happens:
- Coils must be placed close to each other for maximum effect.
- The magnetic field from one coil passes through the other coil's turns.
- Changing field causes a change in magnetic flux linkage and hence an induced emf (by Faraday's Law).
3. What is the mutual inductance law?
The mutual inductance law states that the induced emf in one coil is proportional to the rate of change of current in the other coil.
Mathematically:
- emf2 = -M (dI1/dt)
- Where M is the mutual inductance coefficient.
- Negative sign shows the direction by Lenz's law.
4. What is the SI unit of mutual inductance?
The SI unit of mutual inductance is Henry (H).
In one Henry, an emf of 1 volt is induced in a coil when the current in the other coil changes at the rate of 1 ampere per second.
5. How do you calculate mutual inductance?
To calculate mutual inductance, use the formula:
- M = (N2 × Φ12) / I1
- Where N2 = turns in second coil, Φ12 = magnetic flux due to current I1 in first coil, all linked with the second coil.
- M also depends on the geometry, orientation, and core material.
6. What is the difference between self and mutual inductance?
Self inductance refers to emf induced in a coil due to change in its own current, while mutual inductance involves emf induced in another coil.
Key differences:
- Self: single coil, symbol L
- Mutual: two coils, symbol M
- Applications: Self - chokes; Mutual - transformers
7. What is the role of the dot convention in mutual inductance circuits?
The dot convention indicates the relative polarity of induced emfs in coupled coils.
Importance:
- If current enters both coils at dotted ends, induced emfs have the same polarity.
- If current enters at one dot and exits at the other, polarities are opposite.
- Ensures correct emf direction in circuit analysis, crucial for JEE/Boards exam problems.
8. Can mutual inductance be negative? What does it indicate?
Yes, mutual inductance can be negative if the induced emf opposes the change in current direction due to coil placement or winding.
Negative M indicates:
- The induced emf acts in an opposite direction to the current change in the primary coil.
- Common in certain winding or dot convention arrangements.
9. What are practical applications of mutual inductance?
Mutual inductance is crucial in many real-world devices.
Examples include:
- Transformers (voltage change in power grids)
- Wireless charging (for phones, electric vehicles)
- Inductive sensors and contactless power transfer
- Radio and communication devices
10. How does coil orientation affect mutual inductance?
The orientation and alignment of coils greatly influence mutual inductance.
Key effects:
- Co-axial, closely spaced coils have maximum mutual inductance.
- If coils are perpendicular, mutual inductance is almost zero.
- Proper alignment is essential in transformer and wireless energy designs.
11. What is the mutual inductance formula for two coils?
The standard mutual inductance formula for two coils is:
- M = (μ0 μr N1 N2 A) / l
- μ0 = permeability of free space
- μr = relative permeability
- N1, N2 = number of turns
- A = area
- l = length of solenoid/core
