Electromagnetic Induction-Revision Notes

 CBSE Class 12 Physics

Revision Notes
Chapter-6
Electromagnetic Induction

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• Magnetic Flux: Magnetic flux through a plane of area dA placed in a uniform magnetic field B is
   ϕ=∫B→.dA→=BA cos θ$\varphi =\int \overrightarrow{B}.d\overrightarrow{A}=BAcos\theta$ where θ$\theta$ is the angle between magnetic field lines and area vector of the surface.
• Dimensions of magnetic flux: [ML2T−2A−1]$\left[M{L}^2{T}^{-2}{A}^{-1}\right]$
• SI unit:- Weber (Wb)
• Faraday’s Law:
  a) First Law: Whenever there is a change in the magnetic flux linked with a circuit with time, an induced emf is produced in the circuit which lasts as long as the change in magnetic flux continues.
  b) Second Law: The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linked with the closed circuit i.e. |e|∝dϕdt$|e|\propto \frac{d\varphi }{dt}$
• Lenz’s Law: The direction of the induced emf or current in the circuit is such that it opposes the cause due to which it is produced i.e. it opposes the change in magnetic flux, so that-
  e=−N(dϕdt)$e=-N\left(\frac{d\varphi }{dt}\right)$ where N is the number of turns in the coil
  Lenz’s law is based on energy conservation.
• Induced Emf due to Linear Motion of a Conducting Rod in a Uniform Magnetic Field:
  The induced emf, e=l→.(v→×B→)$e=\overrightarrow{l}.(\overrightarrow{v}\times \overrightarrow{B})$
  If  l→,v→ andB→$\overrightarrow{l},\overrightarrow{v}and\overrightarrow{B}$ are perpendicular to each other, then e=Bvl$e=Bvl$
  Fleming's Right Hand Rule is used to find the direction of the induced current set up in the conductor.
• Induced EMF due to Rotation of a Conducting Rod in a Uniform Magnetic Field: The induced emf is given by
  E=12BωL2$E=\frac{1}{2}B\omega L^2$ 
  where L is the length of the conducting rod.
• Rotation of Rectangular Coil in a Uniform Magnetic Field:

1. Magnetic flux linked with coil, ϕ=NBAcosθ=NBAcosωt$\varphi =NBA\cos \theta =NBA\cos \omega t$
2. Induced emf in the coil, e=−dϕdt=BANωsinωt=e0sinωt$e=-\frac{d\varphi }{dt}=BAN\omega \sin \omega t=e_0\sin \omega t$
3. The induced current in the coil,  I=eR=NBAωRsinωt=eoRsinωt$I=\frac{e}{R}=\frac{NBA\omega }{R}\sin \omega t=\frac{e_o}{R}\sin \omega t$
4. Both emf and current induced in the coil are alternating.

• Self-Induction and Self Inductance:

1. The phenomenon in which an induced emf is produced by changing the current in a coil is called self induction.
   ϕ∝I or ϕ=LIor L=ϕI$\begin{array}{rl}& \varphi \propto I\text{ or }\varphi =LI\\ & or\text{ L=}\frac{\varphi }{I}\end{array}$NϕB${\varphi }_B$ = LI
    where L is a constant, called self-inductance or coefficient of self – induction.
2. S.I. Unit- Henry (H)
3. Dimension- [ML2T-2A-2]
4. Self inductance of a solenoid, L=μ0N2Al$L=\frac{{\mu }_0{N}^{2}A}{l}$
5. Two coils of self – inductances L1 and L2, placed far away (i.e., without coupling) from each other.
6. For series combination: L=L1+L2.......Ln$L=L_1+L_2.......L_n$
7. For parallel combination: 1L=1L1+1L2+....+1Ln$\frac{1}{L}=\frac{1}{L_1}+\frac{1}{L_2}+....+\frac{1}{L_n}$​

• Mutual Induction and Mutual Inductance:

1. On changing the current in one coil, if the magnetic flux linked with a second coil changes and induced emf is produced in that coil, then this phenomenon is called mutual induction.

   ϕ2∝I1 or ϕ2=MI1${\varphi }_2\propto I_1\text{ or }{\varphi }_2=M{I}_1$

   Or  M=ϕ2I1$M=\frac{{\varphi }_2}{I_1}$

   e2=−dϕ2dt=−MdI1dt$e_2=-\frac{d{\varphi }_2}{dt}=-M\frac{d{I}_1}{dt}$

   M=e2−(dI1/dt)$M=\frac{e_2}{-(d{I}_1/dt)}$  
   Therefore, M12 = M21 = M

2. Mutual inductance two coaxial solenoids:
    M=μ0N1N2Al$M=\frac{{\mu }_0{N}_1{N}_{2}A}{l}$

3. If two coils of self- inductance L1 and L2 are wound over each other, the mutual inductance is, M=kL1L2−−−−√$M=k\sqrt{L_1{L}_2}$ where k is called coupling constant.
4. Mutual inductance for two coils wound in same direction and connected in series L=L1+L2+2M$L=L_1+L_2+2M$
5. Mutual inductance for two coils wound in opposite direction and connected in series L=L1+L2−2M$L=L_1+L_2-2M$
6. Mutual inductance for two coils in parallel L=L1L2−M2L1+L2±2M$L=\frac{L_1{L}_2-M^2}{L_1+L_2\pm 2M}$​

• Energy Stored in an Inductor:
  UB=12LI2max$U_B=\frac{1}{2}L{I^2}_{max}$
• Magnetic Energy Density: UB=B22μ0$U_B=\frac{B^2}{2{\mu }_0}$
• Eddy Current: When a conductor is moved in a magnetic field, induced currents are generated in the whole volume of the conductor. These currents are called eddy currents.
• AC Generator: It is a device by which mechanical energy is converted into electrical energy. It is based on the principle of electromagnetic induction. If a coil of N turns and area A is rotated at ν$\nu$ revolutions per second in a uniform magnetic field B, the motional emf is
  e = NBA (2πν$\pi \nu$) sin (2πν$\pi \nu$t) where we have assumed that at time t = 0s, the coil is perpendicular to the field.