Inductance - Definition, Derivation, Types, Examples
Last Updated : 19 Jul, 2024
Magnetism has a mystical quality about it. Its capacity to change metals like iron, cobalt, and nickel when touched piques children's interest. Repulsion and attraction between the magnetic poles by observing the shape of the magnetic field created by the iron filling surrounding the bar magnet will be learned.
According to physicists, the forces that govern both magnetism and electricity are substantially greater than gravity in electromagnetism.
What is Inductance?
Inductance is an electrical circuit attribute that opposes any change in current in the circuit. Electrical circuits have an intrinsic feature called inductance. Whether desired or not, it will always be found in an electrical circuit. The inductance of a straight wire carrying electricity with no iron element in the circuit will be lower. Because the inductance of an electrical circuit opposes any change in current in the circuit, it is equivalent to inertia in mechanics.
Magnetic flux that is proportional to the rate of change of the magnetic field is known as induction. The induced EMF across a coil is related to the rate at which the current through it changes. Inductance is the proportionality constant in that relationship. H is the SI unit for inductance (henry). It is denoted by the letter L. The amount of inductance required to produce an EMF of 1 (V) volt in a coil when the current change rate is 1 Henry is defined as 1 H (Henry).
Factors affecting Inductance
The following are some of the factors that influence inductance:
- The inductor's wire has a specific number of turns.
- The material that was used to make the core.
- The core's appearance.
Faraday established the Electromagnetic Induction Law, which states that by altering the magnetic flux, an electromotive force is induced in the circuit. The concept of induction is derived from Faraday's law of electromagnetic induction. The electromotive force generated to counteract a change in current at a specific time interval is known as inductance.
Take a look at a DC source that has the switch turned on. When the switch is turned on, the current flows from zero to a specific value, causing a change in the flow rate. Consider the flux shift caused by current flow. The flux change is measured in terms of time, as follows:
dφ/dt
Use Faraday's law of electromagnetic induction to solve the problem.
E = N(dϕ/dt)
Where, N is the coil's number of turns, and E is the induced EMF across the coil.
Write the above equation as follows using Lenz's law:
E = -N(dϕ/dt)
For computing the value of inductance, the previous equation is adjusted.
E = -N(dϕ/dt)
∴ E = -L(di/dt)
N = dΦ = L di
NΦ = Li
Therefore,
Li = NΦ = NBA
Where, B denotes the flux density and A denotes the coil area.
Hl = Ni
Where H denotes the magnetic flux's magnetizing force.
B = μH
Li = NBA
L = NBA/i = N2BA/Ni
N2BA/Hl = N2μHA/Hl
L = μN2A/l = μN2πr2/l
Types of Inductance
There are two types of inductance. They are self-induction and mutual induction. Let's learn about them in more detail with proper definitions,
Self Induction
The magnetic flux associated with a coil or circuit changes anytime the electric current running through it changes. As a result, an emf is induced in the coil or circuit, which opposes the change that creates it, according to Faraday's laws of electromagnetic induction. This phenomenon is known as 'self-induction,' and the induced emf is referred to as back emf, while the current created in the coil is referred to as induced current.
- Coefficient of self-induction: The current is proportional to the number of flux linkages with the coil, i.e., Nϕ is directly proportional, or Nϕ = Li (N is the number of turns in coil and Nϕ - total flux linkage). Hence The coefficient of self-induction is L = (Nϕ/i).
- If i = 1amp, N = 1 then, L = ϕ i.e. When the current in a coil is 1 amp, the coefficient of self-induction is equal to the flux associated with the coil.
- Faraday's second law induced emf e = -N(dϕ/dt). Which gives e = -L(di/dt); If di/dt = amp/sec then |e| = L. When the rate of change of current in the coil is unity, the coefficient of self induction is equal to the emf induced in the coil.
- Units and dimensional formula of 'L' : It's S.I. unit, weber/Amp = (Tesla × m2)/ Amp = (N × m)/Amp2 = Joule/Amp2 = (Coulomb × volt)/Amp2 = (volt × sec)/amp = (ohm × sec).
But practical unit is henry (H). It's dimensional formula [L] = [ML2T-2A-2]
- Dependence of self-inductance (L): 'L' is determined by the number of turns (N), the area of cross section (A), and the permeability of the medium, not by the current flowing or changing, but by the number of turns (N), the area of cross-section (A), and the permeability of the medium (μ). 'L' does not play a role in the circuit until there is a steady current running through it. Only when there is a change in current does 'L' enter the picture.
- The magnetic potential energy of inductor: In order to create a continuous current in the circuit, the source emf must work against the coil's self-inductance, and any energy expanded for this work is stored in the coil's magnetic field, which is referred to as magnetic potential energy (U).
U = 1/2 (Li)i = Nϕi/2
The various formulae for L
- Circular coil, L = μ0πN2r/2
- Solenoid, L = μ0N2r/l = μ0n2Al
- Toroid, L = μ0N2r/2
- Square coil, L = 2√2μ0N2a/π
Mutual Induction
When the current going through a coil or circuit varies, so does the magnetic flux coupled to a neighboring coil or circuit. As a result, an emf will be induced in the next coil or circuit. Mutual induction is the term for this occurrence.
- Coefficient of mutual induction: N2ϕ2 is the total flux linked with the secondary due to current in the primary, and N2ϕ2 is directly proportional to i1 = N2ϕ2 = Mi1, where N1 is the number of turns in the primary, N2 is the number of turns in the secondary, ϕ2 is the flux linked with each turn of the secondary, i1 is the current flowing through the primary, and M is the mutual inductance coefficient.
- According to Faraday's second law emf induces in secondary e2=-N(dϕ2/dt); e2=-M(di1/dt)
- If di1/dt = 1Amp/sec then |e2|=M. When the rate of change of current in the main coil is unity, the mutual induction coefficient is equal to the emf induced in the secondary coil.
- Units and dimensional formula of 'M': It's S.I. unit, weber/Amp = (Tesla × m2)/ Amp = (N × m)/Amp2 = Joule/Amp2 = (Coulomb × volt)/Amp2 = (volt × sec)/amp = (ohm × sec).
But practical unit is henry (H). It's dimensional formula [M] = [ML2T-2A-2].
- Dependence of mutual inductance:
- Both coils have the same number of turns (N1, N2).
- Both coils' self-inductance coefficients (L1, L2).
- Coils cross-sectional area.
- The nature of the material on which two coils are coiled or the magnetic permeability of the medium between the coils (μr).
- Two coils are separated by this distance. (As d grows larger, M shrinks.)
- Orientation of main and secondary coils (no flux relation M=0 for 90 degree orientation).
- Between the primary and secondary coils, there is a 'K' coupling factor.
- Relation between M, L1, and L2: For two magnetically coupled coils M = K √L1L2, where k - coefficient of coupling or coupling factor which is defined as,
K = Magnetic flux linked in secondary / Magnetic flux linked in primary
0 ≤ K ≤ 1
The various formulae for M
- Two concentric coplanar circular coils, M = πμ0N1N2r2/2R
- Two Solenoids, M = μ0N1N2A/l
- Two concentric coplanar square coils, M = μ02√2N1N2l2/πL
Combination of Inductance
Series
If two mutually inducing self-inductance coils L1 and L2 are connected in series and separated by a large enough distance that mutual induction between them is insignificant, then net self-inductance Ls = L1 + L2.
When they're near together, the net inductance is Ls = L1 + L2 ± 2M.
Parallel
When two mutually inducting self-inductance coils L1 and L2 are linked in parallel and separated by a large distance, the net inductance L is 1/Lp = 1/L1 + 1/L2.
∴ Lp = L1L2/L1 + L2
When they are in close proximity to one another,
Lp = L1L2 - M2/L1 + L2 ± 2M
Self vs Mutual Inductance
| Self Induction | Mutual Induction |
| The coil's self-inductance is a property of the coil. | The characteristic of a pair of coils is mutual inductance. |
| When the main current in the coil declines, the induced current resists the decay of current in the coil. | When the main current in the coil declines, the induced current created in the nearby coil opposes the decay of the current in the coil. |
| When the coil's primary current grows, the induced current opposes the expansion of current in the coil. | When the coil's primary current grows, the induced current created in the adjoining coil opposes the coil's current development. |
Things to Keep in Mind
- Magnetic flux that is proportional to the rate of change of the magnetic field is known as induction.
- The induced EMF across a coil is related to the rate at which the current through it changes.
- Inductance is the proportionality constant in that relationship.
- H is the SI unit for inductance (henry). It is denoted by the letter L.
- Faraday's law of electromagnetic induction is an electromagnetism fundamental law that describes how a magnetic field interacts with an electric circuit to produce an electromotive force (EMF).
- Faraday's law is the name for this phenomena, which is known as electromagnetic induction.
- Mutual inductance is a property of a pair of conductors, whereas Self Inductance is a property of the conductors individually.
Sample Problems on Inductance
Question 1: Three coils are wired together in a series. Each coil has an inductance of 5H, 4H, and 6H, respectively. Calculate the inductance equivalent.
Solution:
Given: L1 = 5H, L2 = 4H, L3 = 6H
The series inductance all sum as
L = L1 + L2 + L3
∴ L = 5 + 4 + 6
∴ L = 15H
Question 2: What factors have an impact on inductance?
Answer:
The following are some of the factors that influence inductance:
- The inductor's wire has a specific number of turns.
- The material that was used to make the core.
- The core's appearance.
Faraday established the Electromagnetic Induction Law, which states that by altering the magnetic flux, an electromotive force is induced in the circuit. The concept of induction is derived from Faraday's law of electromagnetic induction. The electromotive force generated to counteract a change in current at a specific time interval is known as inductance.
Question 3: Define a coil's self-inductance. Establish a S.I. unit for it.
Answer:
The property of a coil that opposes the growth or decay of the current flowing through it is known as self-induction.
Henry is the SI unit of self-inductance (H).
Question 4: Consider a 500-turn solenoid coiled on an iron core with a relative permeability of 800. The solenoid's length is 40 cm, and its radius is 3 cm. The current changes from 0 to 3 A. Calculate the average induced emf for this change in current at 0.4 second intervals.
Solution:
N = 500 turns, μr = 800, Length = 40 cm = 0.4 m
Radius, r = 3 cm = 0.03 m
Change in current, di = 3 – 0 = 3 A
Change in time, dt = 0.4 sec
Self-inductance is given as
L = μN2Al = μ0μrN2πr2/l
∴ L = (4)(3.14)(10-7)(800)(5002)(3.14)(3 × 10-2)2/0.4
∴ L = 1.77 H
ε = L di/dt = 1.77 × 3/0.4
∴ ε = 13.275 V
Question 5: Explain Combination of Inductance.
Answer:
If two mutually inducting self inductance coils L1 and L2 are connected in series and separated by a large enough distance that mutual induction between them is insignificant, then net self inductance Ls = L1 + L2.
When they're near together, the net inductance is Ls = L1 + L2 ± 2M.
When two mutually inducting self-inductance coils L1 and L2 are linked in parallel and separated by a large distance, the net inductance L is 1/Lp = 1/L1 + 1/L2.
∴ Lp = L1L2/L1+L2
When they are in close proximity to one another,
Lp = L1L2 - M2/L1 + L2 ± 2M
Question 6: Write the difference between Self Inductance and Mutual Inductance.
Answer:
| Self Induction | Mutual Induction |
| The coil's self inductance is a property of the coil. | The characteristic of a pair of coils is mutual inductance. |
| When the main current in the coil declines, the induced current resists the decay of current in the coil. | When the main current in the coil declines, the induced current created in the nearby coil opposes the decay of the current in the coil. |
| When the coil's primary current grows, the induced current opposes the expansion of current in the coil. | When the coil's primary current grows, the induced current created in the adjoining coil opposes the coil's current development. |
Question 7: The inductance of a coil is 6 H, and the supply frequency is 70 Hz. What is the reactance?
Solution:
Given: L = 6H, f = 70Hz
Solution:
X = 2πfL
X = 2 × 3.14 × 70 × 6
X = 2637.6 Ω
Worksheet: Inducatance
Problem 1: Calculate the energy stored in a 5 H inductor when a current of 3 A is flowing through it.
Problem 2: A coil has 500 turns and a magnetic flux of 0.02 Wb is linked with it when a current of 2 A flows through it. Calculate the inductance of the coil.
Problem 3: In an RL circuit, the resistor has a resistance of 10 Ω and the inductor has an inductance of 2 H. Calculate the time constant of the circuit.
Problem 4: A current in a coil changes from 3 A to 1 A in 0.5 seconds, inducing an EMF of 4 V. Calculate the inductance of the coil.
Problem 5: Two coils are placed close to each other, and a change in current of 2 A in one coil induces a voltage of 3 V in the other coil. If the change in current occurs over 0.01 seconds, calculate the mutual inductance between the coils.
Problem 6: What is inductance? Describe it in your own words.
Problem 7: Two inductors, 3 H and 6 H, are connected in series. Calculate the total inductance of the combination.
Problem 8: Two inductors, 3 H and 6 H, are connected in parallel. Calculate the total inductance of the combination.
Problem 9: In an AC circuit, an inductor of 4 H is connected to a 50 Hz supply. Calculate the inductive reactance of the inductor.
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Series LCR CircuitsIn contrast to direct current (DC), which travels solely in one direction, Alternating Current (AC) is an electric current that occasionally reverses direction and alters its magnitude constantly over time. Alternating current is the type of electricity that is delivered to companies and homes, and
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Power Factor in AC circuitThe power factor is determined by the cosine of the phase angle between voltage and current. In AC circuits, the phase angle between voltage and current is aligned, or in other words, zero. But, practically there exists some phase difference between voltage and current. The value of the power factor
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TransformerA transformer is the simplest device that is used to transfer electrical energy from one alternating-current circuit to another circuit or multiple circuits, through the process of electromagnetic induction. A transformer works on the principle of electromagnetic induction to step up or step down th
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CHAPTER 8 - ELECTROMAGNETIC WAVES
CHAPTER 9 - RAY OPTICS AND OPTICAL INSTRUMENTS