Class 12 Physics Electromagnetic Induction Self-Inductance

Self-Inductance

  • There is one coil in which there is change in the flux in that coil and because of that flux change.
  • Current is induced in the same coil.
  • Current tries to oppose the change in the flux.
  • Consider a closed circuit,as a result the current will flow through the coil,therefore flux increases as a result current is induced in the coil.
  • This induced current will oppose the growth of current.
  • Suppose there is N number of turns in the coil.Therefore flux linkage of the coil N Φ∝ I.
  • N Φ =LI where L=constant of proportionality and is known as self-inductance.
  • Therefore Self-inductance will describe about the ratio of magnetic flux to the current it induces.
  • Induced Emf e=-(d Φ/dt) By faraday’s Lenz’s law
  • Therefore e=-d/dt [LI]
  • e= -L dI/dt Where I=current flows through the coil.
  • This Emf will oppose the change in I.

 

Self-inductance of a long solenoid

  • Long solenoid is the one whose length(length) is very large as compared to radius(r) of the solenoid.(l>>r)
  • Using B=μ0nI
  • =(μ0NI)/length where N= total number of turns, n= number of turns per unit length.
  • Case 1:- core of the solenoid has air.
    • Flux=NBA
  • =NA((μ0NI)/length).
  • =>Φ =(μ0N2IA)/length (equation(1))
  • Also total flux=LI (equation(2))
  • From (1) and (2)
  • => LI =(μ0N2IA)/length
  • =>L = (μ0N2A)/length
  • Case 2:- core of the solenoid is made of material which has permeability μr.
    • L=(μr N2A)/length
    • =>L= (μ0 μr N2 A)/length

 

Problem:- A long solenoid with 15 turns per cm has a small loop of area 2.0 placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0A in 0.1 s, what is the induced emf in the loop while the current is changing?

Answer: - Number of turns on the solenoid = 15 turns/cm = 1500 turns/m

Number of turns per unit length, n = 1500 turns

The solenoid has a small loop of area, A =2.0 cm2= 2 × 10-4m2

Current carried by the solenoid changes from 2 A to 4 A.

Therefore change in current in the solenoid, di = 4 – 2 = 2 A

Change in time, dt = 0.1 s

Induced emf in the solenoid is given by Faraday’s law as:

e= (dΦ/dt)   … (i)

Where, Φ= Induced flux through the small loop

= BA … (ii)   Where B = Magnetic field

= (μ0ni) … (iii)

μ0 = Permeability of free space

=4nx10-7H/m

Hence, equation (i) reduces to:

e= (d/dt) (di/dt)

=Aμ0n x (di/dt)

=2x10-4x4x3.14x10-7x 1500 x (2/0.1)

=7.54 x 10-6V

Hence, the induced voltage in the loop is 7.54x10-6V

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