## Electromagnetic Induction

# Faradayâ€™s Law of Induction, Lenzâ€™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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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)