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Inductance | Physics Class 12 - NEET PDF Download

What is Inductance?

  • Inductance is like a resistance to changes in electric current in a wire. We measure it in Henrys, where 1 Henry means it takes 1 volt to make a change in current of 1 Ampere per second. 
  • When electricity moves through a wire, it makes a magnetic field. If the current changes, this field changes too, which creates a kind of electric push-back. This push-back is what we call inductance. We use a component called an inductor to add inductance to a circuit.

An InductorAn Inductor

  • Inductance is classified into two types as:
  • Self Inductance
  • Mutual Inductance

Self induction

  • Self induction is induction of emf in a coil due to its own current change. Total flux N Inductance | Physics Class 12 - NEETpassing through a coil due to its own current is proportional to the current and is given as N Inductance | Physics Class 12 - NEET = L i where L is called coefficient of self induction or inductance. 
  • The inductance L is purely a geometrical property i.e., we can tell the inductance value even if a coil is not connected in a circuit. Inductance depends on the shape and size of the loop and the number of turns it has.

If current in the coil changes by ΔI in a time interval Δt, the average emf induced in the coil is given as

ε = Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

The instantaneous emf is given as

ε = -Inductance | Physics Class 12 - NEET = - Inductance | Physics Class 12 - NEET

S.I unit of inductance is wb/amp or Henry (H)

L - self inductance is +ve Quantity.

L depends on : 

(1) Geometry of loop

(2) Medium in which it is kept. 

L does not depend upon current. L is a scalar Quantity.

Question for Inductance
Try yourself:What is the unit of measurement for inductance?
View Solution

Self Inductance of solenoid

Inductance | Physics Class 12 - NEETLet the volume of the solenoid be V, the number of turns per unit length be n. Let a current I be flowing in the solenoid. Magnetic field in the solenoid is given as B = μ0 nl. The magnetic flux through one turn of solenoid f = μ0 n l A.

The total magnetic flux through the solenoid = N Inductance | Physics Class 12 - NEET

= N μ0 n l A

= μ0 n2 l A l

Therefore, L = μ0 n2 l A = μ0 n2 V

Ф= μ0 n i pr2 (n l)

L =  Ф/i = μ0 n2 πr2 l

Inductance per unit volume = μ0 n2

Ex. The current in a coil of self-inductance L = 2H is increasing according to the law i = 2 sin t2. Find the amount of energy spent during the period when the current changes from 0 to 2 ampere.

Sol. Let the current be 2 amp at t = τ

Then 2 = 2 sin τ2 ⇒ τ = Inductance | Physics Class 12 - NEET

When the instantaneous current is i, the self induced emf is L Inductance | Physics Class 12 - NEET. If the amount of charge that is displaced in time dt is dθ, then the elementary work done

= Inductance | Physics Class 12 - NEET

W = Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

W = Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

Let θ = 2t2

Differentiating dθ = 4t dt

Therefore, W = Inductance | Physics Class 12 - NEET

= L (- cosθ) = - L cos 2t2

W = Inductance | Physics Class 12 - NEET

= 2 L = 2 × 2 = 4 joule

Inductor

It is represented by

Inductance | Physics Class 12 - NEET

electrical equivalence of loop

Inductance | Physics Class 12 - NEET ⇒ Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET

If current i through the inductor is increasing the induced emf will oppose the increase in current and hence will be opposite to the current. If current i through the inductor is decreasing the induced emf will oppose the decrease in current and hence will be in the direction of the current.

Inductance | Physics Class 12 - NEET

Over all result

Inductance | Physics Class 12 - NEET

VA - L Inductance | Physics Class 12 - NEET = VB

Note

Inductance | Physics Class 12 - NEET If there is a resistance in the inductor (resistance of the coil of inductor) then :

Inductance | Physics Class 12 - NEET Inductance | Physics Class 12 - NEET

Ex. A B is a part of circuit. Find the potential difference vA - vB if

Inductance | Physics Class 12 - NEET

(i) current i = 2A and is constant

(ii) current i = 2A and is increasing at the rate of 1 amp/sec.

(iii) current i = 2A and is decreasing at the rate 1 amp/sec.

Sol. Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

writing KVL from A to B

VA - 1Inductance | Physics Class 12 - NEET - 5 - 2 i = VB

(i) Put i = 2, Inductance | Physics Class 12 - NEET = 0

VA - 5 - 4 = VB    Therefore, VA - VB = 9 volt

(ii) Put i = 2, Inductance | Physics Class 12 - NEET = 1 ;

VA - 1 - 5 - 4 = VB or VA - VB = 10 V0

(iii) Put i = 2, Inductance | Physics Class 12 - NEET = - 1

VA + 1 - 5 - 2 × 2 = VB

VA = 8 volt

Ex. Find current i, i1 and i2 in the following circuit.

Inductance | Physics Class 12 - NEET

Sol. at t = 0

i = i2 = Inductance | Physics Class 12 - NEET and i1 = 0

at t = ∞

Inductance | Physics Class 12 - NEET

⇒ i1 = i2 = Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

Energy stored in an inductor 

If current in an inductor at an instant is i and is increasing at the rate di/dt, the induced emf will oppose the current. Its behaviour is shown in the figure.

Inductance | Physics Class 12 - NEET

Power consumed by the inductor = i L Inductance | Physics Class 12 - NEET

Energy consumed in dt time = i L Inductance | Physics Class 12 - NEET dt

Therefore, total energy consumed as the current increases from 0 to I = Inductance | Physics Class 12 - NEET

= Inductance | Physics Class 12 - NEET ⇒ U = Inductance | Physics Class 12 - NEET

Note :

Inductance | Physics Class 12 - NEET This energy is stored in the magnetic field with energy density

Inductance | Physics Class 12 - NEET

Total energy U = Inductance | Physics Class 12 - NEET

Ex. Find out the energy per unit length ratio inside the solid long wire having current density J.

Inductance | Physics Class 12 - NEET

Sol. Take a ring of radius r and thickness dr as an element inside the wire

Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET        

using B = Inductance | Physics Class 12 - NEET    Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET ⇒ Inductance | Physics Class 12 - NEET ⇒ Inductance | Physics Class 12 - NEET

Mutual Inductance

  • Consider two coils P and S placed close to each other as shown in the figure. When the current passing through a coil increases or decreases, the magnetic flux linked with the other coil also changes and an induced e.m.f. is developed in it. 
  • This phenomenon is known as mutual induction. This coil in which current is passed is known as primary and the other in which e.m.f. is developed is called as secondary.

Let the current through the primary coil at any instant be i1. Then the magnetic flux Inductance | Physics Class 12 - NEET in the secondary at any time will be proportional to i1 i.e., Inductance | Physics Class 12 - NEET is directly proportional to i1

Therefore the induced e.m.f. in secondary

when i1 changes is given by

Inductance | Physics Class 12 - NEET i.e., Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET ⇒ Inductance | Physics Class 12 - NEET = M i1            Inductance | Physics Class 12 - NEET

where M is the constant of proportionality and is known as mutual inductance of two coils. It is defined as the e.m.f. induced in the secondary coil by unit rate of change of current in the primary coil. The unit of mutual inductance is henry (H).

Question for Inductance
Try yourself:
What is the definition of mutual inductance?
View Solution

Mutual Inductance of a Pair of Solenoids one Surrounding the other coil

Figure shows a coil of N2 turns and radius R2 surrounding a long solenoid of length l1, radius R1 and number of turns N1.

Inductance | Physics Class 12 - NEET

To calculate mutual inductance M between them, let us assume a current i1 through the inner solenoid S1

There is no magnetic field outside the solenoid and the field inside has magnitude,

Inductance | Physics Class 12 - NEET

and is directed parallel to the solenoid's axis. The magnetic flux Inductance | Physics Class 12 - NEET through the surrounding coil is, therefore,

Inductance | Physics Class 12 - NEET

Now, Inductance | Physics Class 12 - NEET

Notice that M is independent of the radius R2 of the surrounding coil. This is because solenoid's magnetic field is confined to its interior.

Ex.Find the mutual inductance of two concentric coils of radii a1 and a(a1 << a2) if the planes of coils are same.

Inductance | Physics Class 12 - NEET

Sol. Let a current i flow in coil of radius a2.

Magnetic field at the centre of coil =

or M i = Inductance | Physics Class 12 - NEET or Inductance | Physics Class 12 - NEET

Ex. Solve the above question, if the planes of coil are perpendicular.

Sol. Let a current i flow in the coil of radius a1. The magnetic field at the centre of this coil will now be parallel to the plane of smaller coil and hence no flux will pass through it, hence M = 0

Ex. Solve the above problem if the planes of coils make θ angle with each other.

Sol. If i current flows in the larger coil, the magnetic field produced at the centre will be perpendicular to the plane of larger coil.

Now the area vector of smaller coil which is perpendicular to the plane of smaller coil will make an angle θ with the magnetic field.

Thus flux = Inductance | Physics Class 12 - NEET. πa12 cos θ

or M = Inductance | Physics Class 12 - NEET

Ex. Find the mutual inductance between two rectangular loops, shown in figure.

Inductance | Physics Class 12 - NEET

Sol. Let current i flow in the loop having ∞-by long sides. Consider a segment of width dx at a distance x as shown flux through the regent

Inductance | Physics Class 12 - NEET

dInductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

= Inductance | Physics Class 12 - NEET

Ex. Figure shows two concentric coplanar coils with radii a and b (a << b). A current i = 2t flows in the smaller loop. Neglecting self inductance of larger loop

Inductance | Physics Class 12 - NEET

(a) Find the mutual inductance of the two coils

(b) Find the emf induced in the larger coil

(c) If the resistance of the larger loop is R find the current in it as a function of time

Sol. (a) To find mutual inductance, it does not matter in which coil we consider current and in which flux is calculated (Reciprocity theorem) Let current i be flowing in the larger coil. Magnetic field at the centre = Inductance | Physics Class 12 - NEET.

flux through the smaller coil = Inductance | Physics Class 12 - NEET

Therefore, M = Inductance | Physics Class 12 - NEET

(ii) |emf induced in larger coil| = Inductance | Physics Class 12 - NEET

= Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

(iii) current in the larger coil = Inductance | Physics Class 12 - NEET

Ex. If the current in the inner loop changes according to i = 2t2 then, find the current in the capacitor as a function of time.

Inductance | Physics Class 12 - NEET

Sol. M = Inductance | Physics Class 12 - NEET

|emf induced in larger coil| = Inductance | Physics Class 12 - NEET

e = Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET    Inductance | Physics Class 12 - NEET

Applying KVL : -

+e - Inductance | Physics Class 12 - NEET - iR = 0 ⇒ Inductance | Physics Class 12 - NEET - -iR = 0

differentiate wrt time : Inductance | Physics Class 12 - NEET - Inductance | Physics Class 12 - NEET - Inductance | Physics Class 12 - NEETon solving it i = Inductance | Physics Class 12 - NEET

Series Combination of Inductors

Inductance | Physics Class 12 - NEET = Inductance | Physics Class 12 - NEET

Therefore, V = V1 + V2

Inductance | Physics Class 12 - NEET

L = L1 + L2 +.................

Parallel Combination of inductor

i = i1 + i2Inductance | Physics Class 12 - NEET     Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET

Inductance | Physics Class 12 - NEET

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FAQs on Inductance - Physics Class 12 - NEET

1. What is self-inductance?
Ans. Self-inductance is the property of an electrical circuit that causes a change in current in the circuit to induce a voltage in the same circuit.
2. How is self-inductance of a solenoid defined?
Ans. The self-inductance of a solenoid is a measure of the ability of the solenoid to induce a voltage in itself when the current through it changes.
3. What is the energy stored in an inductor?
Ans. The energy stored in an inductor is given by the formula W = (1/2) * L * I^2, where L is the inductance of the inductor and I is the current flowing through it.
4. What is mutual inductance?
Ans. Mutual inductance is the property of two circuits where a change in current in one circuit induces a voltage in the other circuit.
5. How is the mutual inductance of a pair of solenoids, where one surrounds the other coil, calculated?
Ans. The mutual inductance of a pair of solenoids is calculated using the formula M = μ0 * N1 * N2 * A / l, where μ0 is the permeability of free space, N1 and N2 are the number of turns in each solenoid, A is the cross-sectional area, and l is the length of the solenoids.
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