## Electromagnetic Induction

# Faraday’s Law of Induction, Lenz’s Law and Conservation of Energy

- If the current flowing in the primary coil is ‘ip’, ‘φ
_{s}’ is the magnetic flux of the secondary coil, then φ_{s}∝ i_{p}⇒ φ_{s}= Mi_{p}⇒ \tt M = \frac{\phi_{s}}{i_{p}} - Induced emf generated in secondary coil is \tt e = - \frac{d \phi}{dt} = - M \left(\frac{di_{p}}{dt}\right)

\tt M = \frac{e}{\left(-di_{p} / dt \right)} - Dimensional formula of mutual inductance is [ML
^{2}T^{−2}A^{−2}] - Two solenoids with numbers of turns per unit length n
_{1}and n_{2}, with coil 1 inside, coil 2 co-axially placed. Then \tt M = \mu_{0} n_{1}n_{2} \pi r_{1}^{2} l - If a magnetic material of relative permeability μ
_{r}fills the space inside the solenoid, then \tt M = \mu_{r} \mu_{0} n_{1}n_{2} \pi r_{1}^{2} l - Mutual inductance for coil 1 due to 2 is equal to mutual inductance for the coil 2 due to 1 ⇒ M
_{12}= M_{21} - Two circular coils, one with very small radius r
_{1}and another with very large radius r_{2}are placed co-axially with their centres coinciding, then \tt M = \frac{\mu_{0} \pi r^{2}_{1}}{2r_{2}} - If ‘I’ is the current flowing through the coil and ‘φ’ is the magnetic flux linked with the coil, then φ ∝ i ⇒ φ = Li⇒ \tt L = \frac{\phi}{i}

L = co-efficient of self-induction

- Self-induced emf, \tt e = \frac{- d \phi}{dt} = - L \frac{di}{dt}
- A coil with high self-inductance is known as INDUCTOR.
- In case of a long solenoid, where the core consists of a magnetic material of relative permeability μ
_{r}, then L = μ_{r}μ_{0}n^{2}Al - Energy in a current carrying coil is stored in the form of magnetic field and it is given by \tt U = \frac{1}{2} Li^{2}
- Relation between self-inductance and mutual inductance is given as \tt M = K \sqrt{L_{1}L_{2}} (K = coupling factor), K ≤ 1
- Coupling factor, K = 1 If the coils are wound one over the other
- Inductances are added in series or parallel L
_{series}= L_{1}+ L_{2}+ L_{3}+ …… - \tt \frac{1}{L_{parallel}} = \frac{1}{L_{1}} + \frac{1}{L_{2}} + \frac{1}{L_{3}} + ....
- An AC GENERATOR converts mechanical energy into electrical energy.
- The method to induce an emf in a loop is through a change in the loop’s orientation. This is the PRINCIPLE OF SIMPLE AC GENERATOR
- The induced emf for the rotating coil of N turns is,

\tt \varepsilon = - \frac{d \phi_{B}}{dt} = - \frac{d}{dt} (NBA cos ωt)

∴ ε = NBA ω sin ωt

- The magnitude of induced emf is

ε = NBA ω sin ωt = ε_{0}sinωt

ε_{0} = peak value of the emf

- ‘i’ is the current in the circuit at any instant ‘t’ in an L-R circuit with DC current

\tt i = i_{0} \left\{{1 - e^{-t / \lambda}}\right\} , \lambda = \frac{L}{R}, inductive time constant.

- At t = λ, i = i
_{0}\tt \left(1 - \frac{1}{e}\right) = 0.63 i_{0} - The INDUCTIVE TIME CONSTANT of a circuit is defined as the time in which the current rises from zero to 63% of its final value.
- ‘i’ is the current in the circuit at any instant in an L-R circuit with DC current and \tt i = i_{0} e^{-t/ \lambda}
- At t = λ, \tt i = \frac{i_{0}}{e} = 0.37i_{0}
- The inductive time constant (λ) is defined as the time interval during which the current decays to 37% of the maximum current
- The charge present on the plates opposes further introduction of charge \tt E - \frac{q}{c} = Ri \Rightarrow E - \frac{q}{c} = R \frac{dq}{dt}
- λ = CR is called CAPACITIVE TIME CONSTANT
- \tt q = q_{0} \left(1 - \frac{1}{e} = 0.63 q_{0}\right)
- The CAPACITIVE TIME CONSTANT is the time in which the charge on the plates of the capacitor becomes 0.63 q
_{0}. - If the charge slowly reduces to zero after infinite time then \tt \frac{-q}{c} = Ri \Rightarrow \frac{-q}{c} = R \frac{dq}{dt}
- At t = λ, \tt q = \frac{q_{0}}{e} = 0.37 q_{0}

### View the Topic in this video From 26:16 To 55:11

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1. The induced emf is given by rate of change of magnetic flux linked with the circuit, i.e., e = -\frac{d\phi}{dt}. For N turns e = -\frac{N d\phi}{dt}; Negative sign indicates that induced emf (e) opposes the change of flux.

2. When Φ changes, the average induced emf

e = -N \frac{d\phi}{dt} = - \frac{N(\phi_{2} - \phi_{1})}{\Delta t} = - \frac{NA(B_{2} - B_{1}) cos \theta}{\Delta t} = - \frac{NBA(\cos \theta_{2} - \cos \theta_{1})}{\Delta t}

3. Φ = NBA cos θ = NBA cos ωt, where ω is the angular velocity e = -\frac{d \phi}{dt} = - \frac{d}{dt}(NBA \ \cos \omega t)