Electromagnetic Induction PDF

# ELECTRO MAGNETIC INDUCTION ## Magnetic Flux: - If we consider a plane perpendicular to a uniform magnetic field, then the product of the magnitude of the field and the area of the plane is called magnetic flux. - $\phi$ = B.A - SI unit of magnetic flux is 'weber'. ## Faraday's Law of Electromagn...

# ELECTRO MAGNETIC INDUCTION ## Magnetic Flux: - If we consider a plane perpendicular to a uniform magnetic field, then the product of the magnitude of the field and the area of the plane is called magnetic flux. - $\phi$ = B.A - SI unit of magnetic flux is 'weber'. ## Faraday's Law of Electromagnetic Induction **First law:** When the magnetic flux through a circuit is changing; an induced emf is set up in the circuit whose magnitude is equal to the negative rate of change of magnetic flux. - e = - $\frac{ΔΦB} {Δt}$ - As Δt → 0, e = - $\frac{dΦB}{dt}$ - If N turns, e = - $\frac{d(NPB)}{dt}$ = -N $\frac{dPB}{dt}$ **Second Law:** the direction of the induced emf, or the current, in any circuit is such as to oppose the cause that produces it. This is also known as Lenz's law. ## Induced current: - If in a coil of N turns the rate of change of magnetic flux be $\frac{dPB}{dt}$, then induced emf = -N $\frac{dPB}{dt}$ - If the coil be closed and the total resistance of the circuit be R, the induced current I = $\frac{N}{R}$ $\frac{ΔΦB}{Δt}$ ## Motional emf: - Suppose a thin conducting rod length l is situated in a uniform magnetic field B perpendicular (⊥) to the plane of paper directed downward. The field B has been represented by crosses, rod moves with const. velocity v toward right. - Fm = q v B (fact on charges) - Lorentz force: - positive charges directed towards upward - negative charge directed towards ­­­­­­­downward - J becomes positively charged and K becomes negatively charged - Hence, electric potential difference V is induced between the ends of the rod, due to which an electric field E is equated with in the rod. - E = $\frac{V}{l}$ - Force by E, Fe = qE - At equilibrium, Fe = Fm - qE = q v B - E = Bv - The induced potential difference between the ends of the rod called motional emf. - V = Bvl - Fig-2 - **Induced Current:** - _Fig-2_ - I = $\frac{e}{R}$ - I = $\frac{Bvl}{R}$ - Due to presence of magnetic field, force on the conductor JK will be - F = I lB = $\frac{B^2 l^2 v}{R}$ ## Motional emf in Rotating a Conducting Rod in a uniform magnetic field: - |e| = $\frac{1}{2}$ Bωl^2 - Fig-3 - where - - l - length of rod - ω angular velocity of Rod - B perpendicular magnetic field ## Self Induction - The phenomenon of electromagnetic induction in which, on changing the current in a coil, an opposing induced emf is setup in that very coil is called self-induction. ## Coefficient of Self Induction - Let us consider a coil of N turns carry current I. Let ΦB be the magnetic flux linked with each turn of the coil, then - NΦB α I - NΦB = LI - where L = coefficient of self induction. - L = $\frac{NΦB}{I}$ ## Define coefficient of self Induction: - The coefficient of self induction of coil is equal to the number of flux - linkage with the coil when unit current is flowing through the coal. - By Faraday's law, e = - $\frac{ΔΦB}{Δt} $ = -$Δ(NΦP) $ /Δt = - Δ(LI) / Δt - e = - LΔI / Δt - SI unit of the coefficient of self Inductance is benery (H) - Dimension of [L] = [ML^2T^-2A^-2] ## Self-Inductance of a Long Solenoid - B = μ₀nI, { magnetic field inside solenoid } - magnetic flux through each turn NΦB = Nμ₀nIA = μ₀N^2IA - The self Inductance L = NΦB / I = μ₀N^2A / l - If the solenoid be wound on a 'core of constant permeability μ, the - Total magnet flux = μN^2IA - coefficient of self Inductance - L = μNPA / l { μ = μ₀r } ## Mutual Induction: - If we place two coil near each other and pass electric current in one of them, or change the current already passing through it, or stop the current, then an emf is induced in the second coil. this phenomenone is called mutual inductance. - (ON Key.) ## Coefficient of mutual Inductance: - Let a current of I, ampere flow in the primary coil. Let due to this current, the magnetic flux linked with each turn of the secondary coil be Φ2. - If N2 be the number of turns in secondary coil, then - N2 Φ2 α I - N2 Φ2 = MI - Where'm is constant called 'coefficient of mutual inductance'. - M = N2Φ2 / I - If on changing the current in primary coil, then emf Induced in the secondary coil be e2. - According to Forraday law e2 = -N2ΔΦ2 / Δt = -Δ (N2Φ2) / Δt - e2 = - Δ(MI) / Δt = -MΔI / Δt - M = -e2 / ΔI / Δt - If ΔI = 1 then m = e2 - The coefficient of mutual inductance of two coil is equal to the numerical value of the induced emf in the secondary coil when the rate of change of current in the primary coil is unity. ## Mutual Induction of Two 'Long Coaxial Solenoid - Let us consider a long aircored solenoid P of length l and area of cross section A, having N1 turns., Let a slightly shorter secondary coil S having N2 turns.be wound closely over the primary P. Let I, be the current in the primary. - The B = μ₀nI / l - magnetic flux linketkel with P - Φ2 = B.A = μ₀nI / *l* I A = (1) - When A = πr^2 - By definition, the mutual inductancı of two solenoid is - M = N2Φ2 / I - From (1) - M = μ₀N1N2A / l - If ni be the number turns per unit length in primory, then - n1 = NI / l - M = μ₀n1N2A ## Eddy Current: - When conductor are subjected to changing magnetic flux induced current are produced in them. Their flow patterns resemble swirling eddies in water hence they are called eddy currents. ## Disadvantages of eddy Currents: - Eddy worrents are undesirable as they heat up the core and dissipati electrical energy in the form of heat. ## Methods of reducing eddy currents 1. Holes or slits made in the plate reduces the area available for eddy currents to flow. 2. Laminations- of metal used to make core. The plane lamination must be parallel to the magnetic field so that they cut across. the eddy current paths.