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 [ML²T⁻²A⁻²]
• Two solenoids with numbers of turns per unit length n₁ and n₂, 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 ⇒ M₁₂ = M₂₁
• Two circular coils, one with very small radius r₁ and another with very large radius r₂ 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_rμ₀n²Al
• 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 = L₁ + L₂ + L₃ + ……
• \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 = ε₀sinωt

ε₀ = 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 = i₀ \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 q₀.
• 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}