How to Calculate Inductance for Solenoids and Coils in Physics
Inductance is an important physical quantity in the study of electromagnetism. It describes a material’s or coil’s ability to oppose any changes in electric current flowing through it.
When a current passes through a conductor, it produces a magnetic field, and any variation in that current induces a voltage (or electromotive force) either in the same circuit (self-inductance) or in a nearby circuit (mutual inductance).
What is Inductance?
Inductance is the property of a conducting material that resists the change in electric current, whether caused by internal or external factors. When the electric current through a coil or conductor changes, it creates a changing magnetic field, inducing a voltage according to Faraday’s Law. This induced voltage tries to oppose the change in current.
Mathematically, the dimensional formula for inductance is [ML2T-2I-2]. The unit used to measure inductance is the Henry (H).
Inductance Formulas
Inductance can be calculated using two primary formulas depending on the context:
- General Formula: L = φi / I
L = inductance
φi = magnetic flux (in Weber, Wb)
I = electric current (in Ampere, A) - Coil/Solenoid Formula: L = (μ × N2 × A) / l
μ = permeability of material
N = number of turns
A = cross-sectional area (in m2)
l = length of coil/solenoid (in meters)
| Formula | What it Represents | Variables | SI Unit |
|---|---|---|---|
| L = φi / I | General Inductance | φi = Magnetic flux, I = Current | Henry (H) |
| L = (μ × N2 × A) / l | Inductance of a Coil/Solenoid | μ = Permeability, N = Turns, A = Area, l = Length | H |
Types of Inductance
There are two main types of inductance:
-
Self-Inductance:
This occurs when a changing current in a coil induces a voltage in the same coil. The magnetic field generated by the changing current opposes any change in current.
-
Mutual Inductance:
This occurs when the changing magnetic field from one coil induces a voltage in a nearby coil. The interaction takes place due to the magnetic flux passing through neighboring coils.
Inductance Formula Solved Examples
| Example | Given Values | Solution Steps | Answer |
|---|---|---|---|
| 1. Material with magnetic flux = 0.58, current = 0.9 A | φ = 0.58, I = 0.9 A | L = φ / I = 0.58 / 0.9 | 0.64 H |
| 2. Solenoid: N = 200, A = 27 cm2, l = 56.6 cm, μ = 4π×10-7 | N = 200, A = 27×10-4 m2, l = 56.6×10-2 m | L = (μ × N2 × A) / l | 2.4 × 10-4 H |
| 3. Material with magnetic flux = 0.67, current = 2 A | φ = 0.67, I = 2 A | L = 0.67 / 2 | 0.335 H |
| 4. Material with magnetic flux = 2.8, current = 0.08 A | φ = 2.8, I = 0.08 A | L = 2.8 / 0.08 | 35 H |
Step-by-Step Approach for Inductance Problems
| Step | Description |
|---|---|
| 1 | Identify all given values (such as flux, current, number of turns, area, length, and permeability). |
| 2 | Choose the appropriate formula based on whether the situation is for a general conductor or a coil/solenoid. |
| 3 | Substitute the values into the formula, making sure all units are standard (SI system). |
| 4 | Calculate the answer stepwise and write the final result with units. |
Key Points and Applications
- Inductance is essential in designing inductors, and transformers.
- Used in electromagnetic induction and understanding energy storage in magnetic fields.
- Helps analyze AC circuit behavior and energy conversion.
Further Learning and Practice
- Practice more numerical problems at Inductance Formula Solved Examples.
- For a foundational understanding, see linked topics like LCR circuits.
Continue practicing stepwise solutions and review conceptual explanations to master inductance and related electromagnetic topics. This will strengthen your understanding for competitive exams and for in-depth study in Physics.
FAQs on Inductance Formula Explained with Solved Examples
1. What is the formula of inductance L?
The formula for inductance (L) is given by:
L = NΦ / I
- N = number of turns
- Φ = magnetic flux (in Weber)
- I = current (in Ampere)
For a solenoid, the formula is:
L = (μ₀μrN²A) / l
- μ₀ = permeability of free space
- μr = relative permeability
- N = number of turns
- A = cross-sectional area
- l = length of the solenoid
2. How do you calculate inductance?
To calculate inductance:
- Identify if you have a coil or solenoid
- Use L = NΦ / I for a general coil (where N = number of turns, Φ = magnetic flux, I = current)
- For solenoids, use L = (μ₀μrN²A) / l
- Substitute known values with correct SI units
- Calculate step by step and always check the resulting units (Henry, H)
3. What is the unit of inductance?
The SI unit of inductance is the Henry (H).
- 1 Henry (H) is the inductance when a current change of 1 ampere per second induces an emf of 1 volt.
- Symbol: H
The dimensional formula is [ML2T-2I-2].
4. What is the formula for induced emf?
The induced emf (electromotive force) due to self-induction is given by:
ε = -L (dI/dt)
- ε = induced emf (in Volt, V)
- L = inductance (in Henry, H)
- dI/dt = rate of change of current with time
- The negative sign shows the direction opposes the change in current (Lenz’s Law)
5. What is the difference between self inductance and mutual inductance?
Self inductance is the property of a coil to oppose changes in its own current.
Mutual inductance is the property where a change in current in one coil induces emf in a nearby coil.
Key differences:
- Formula for self-inductance: L = NΦ / I
- Formula for mutual inductance: M = (μ₀μrN₁N₂A) / l
- Application: Self = chokes, inductors; Mutual = transformers, wireless charging
6. How does frequency affect inductive reactance in an AC circuit?
Inductive reactance (XL) increases with frequency.
Formula: XL = 2πfL
- f = frequency (Hz)
- L = inductance (Henry, H)
Higher frequency means greater opposition to AC, so XL is directly proportional to frequency.
7. What are the factors affecting the inductance of a solenoid?
The inductance of a solenoid depends on:
- Number of turns (N): L increases with more turns
- Cross-sectional area (A): Larger area increases L
- Length (l): Longer solenoid reduces L
- Permeability (μ, includes μ₀ and μr): Using materials with higher permeability increases L
The relation is: L = (μ₀μrN²A) / l
8. What is the dimensional formula of inductance?
The dimensional formula of inductance (L) is:
[M L2 T-2 I-2]
- M = mass
- L = length
- T = time
- I = electric current
9. What will happen to the inductance if the length of a solenoid is doubled?
If the length (l) of a solenoid is doubled, its inductance (L) will be halved.
This is because L ∝ 1/l, so increasing l results in a proportional decrease in L, with all other factors constant.
10. How is mutual inductance between two coils calculated?
Mutual inductance (M) between two coils is given by:
M = (μ₀μrN₁N₂A) / l
- N₁, N₂ = number of turns in the two coils
- A = common cross-sectional area
- l = length of the coils
- μ₀ = permeability of free space
- μr = relative permeability of the core material
Use this formula when both coils are wound on the same core and are closely coupled.
11. What is the use of the inductance formula in competitive exams?
The inductance formula is essential in Physics for:
- Solving numerical problems in JEE, NEET, and Board exams
- Deriving relations for coils and circuits
- Calculating energy stored in magnetic fields
- Finding induced emf and current opposition
Mastery of this formula ensures confidence in electromagnetism topics and boosts exam scores.
12. What is the application of self and mutual inductance in real life?
Self-inductance is used in:
- Inductors, chokes, relays, and energy storage in power supplies
Mutual inductance is the principle behind:
- Transformers (voltage conversion)
- Wireless charging systems
- Inductive sensors and communication circuits
