Chapter 6 - Electromagnetic Induction

Faraday did two experiments:

Experiment 1: Coil connected to galvanometer, bar magnet moved near coil. When magnet stationary no deflection, when magnet moves towards coil galvanometer deflects, when magnet moves away opposite deflection, faster motion larger deflection. Relative motion between magnet and coil induces emf.

Experiment 2: Two coils primary connected to battery and secondary to galvanometer placed close. When primary circuit switched on or off or current changed, secondary shows deflection induced emf in secondary due to change current in primary changing flux through secondary.

Magnetic flux φ_B is measure of number of magnetic field lines crossing area. Definition φ_B = B·A = B A cosθ where A is area vector magnitude area direction normal outward, θ angle between B and normal. SI unit Weber Wb: 1 Wb = 1 T·m². Flux depends on B magnitude, area, orientation. Flux can change if B changes, area changes, orientation θ changes.

Example: If B=0.5T area 0.01m² θ=0° φ=0.5×0.01×1=0.005Wb. If θ=90° φ=0.

Faraday's law: Induced emf e equals negative rate of change of magnetic flux: e = -dφ_B/dt. Negative sign represents Lenz law. If coil has N turns total flux linkage N φ_B, e = -N dφ_B/dt.

Lenz law: Direction of induced current is such that it opposes the change causing it. This is conservation of energy. If magnet N pole approaches coil, induced current creates N pole on coil side opposing approach, need to do work against repulsion mechanical work converted to electrical energy. If induced current helped approach, it would accelerate and produce energy without work violation of energy conservation, so Lenz law must oppose change.

Example: N pole approaching coil, coil becomes N pole opposing approach repulsive force, you must push magnet doing work that appears as electrical energy in coil. N pole retreating, coil becomes S pole attracting opposing retreat you must pull doing work.

Induced current I = e/R = -(1/R) dφ_B/dt where R resistance circuit. Charge flown Q = ∫ I dt = -(1/R) Δφ_B depends only on change in flux not rate.

Faraday law is one of Maxwell equations.

When conductor moves in magnetic field, emf induced called motional emf.

Consider rod length l moving with velocity v perpendicular to uniform magnetic field B field into page. Free electrons in rod experience Lorentz force F=q v B perpendicular to v and B, electrons accumulate one end, positive other end, electric field E builds, equilibrium qE=qvB so E=vB. Potential difference e=E l=B l v.

General if angle between v and B is θ, e=B l v sinθ.

If rod slides on U-shaped conductor forming circuit resistance R, current I=e/R=B l v/R flows. Magnetic force on current carrying rod F=I l B=B² l² v/R opposite to motion, so to move rod constant velocity need apply equal opposite force mechanical power P_mech=F v=B² l² v²/R electrical power P_elec=I e=B² l² v²/R equal energy conserved mechanical to electrical.

Fleming's right hand rule gives direction induced current: Stretch thumb, forefinger, middle finger mutually perpendicular thumb direction motion v, forefinger direction magnetic field B, middle finger direction induced current I.

Rotating coil in magnetic field flux φ=N B A cosθ θ=ωt e=-dφ/dt=N B A ω sinωt sinusoidal AC.

When bulk piece of conductor placed in changing magnetic field or moves in magnetic field, induced currents flow in closed circular loops inside conductor called eddy currents. Similar to whirlpools eddies in water. Direction by Lenz law oppose change.

Production: Conductor moving in magnetic field, or magnetic field changing through conductor, flux change induces circular currents in bulk.

Eddy currents produce heat I²R loss, cause energy loss. They oppose motion by Lenz law magnetic braking.

Applications: Magnetic braking trains, electromagnetic damping in galvanometer coil wound on aluminium frame eddy currents oppose motion quickly bring pointer to rest, induction furnace metal heated by eddy currents produced by high frequency AC through coil, electric power meters aluminium disc rotates eddy currents braking, induction cooker.

Undesirable in transformers and motors cores: Changing flux induces eddy currents in iron core causing energy loss heat. Reduced by laminations: Core made of thin sheets laminae insulated from each other stacked, increasing resistance path reducing eddy current magnitude because resistance higher for small loops. Laminated core reduces eddy loss.

Self induction phenomenon where changing current in coil induces emf in same coil itself. Because current produces magnetic field flux through coil itself φ∝I. When I changes flux changes emf induced opposing change inertia.

Flux linkage through coil N φ_B ∝ I so N φ_B = L I where L self inductance constant depends geometry N² A l μ0. Self induced emf e = -d(N φ_B)/dt = -L dI/dt. Negative sign Lenz law opposes change. If current increasing dI/dt positive e negative opposite battery trying decrease current, if current decreasing dI/dt negative e positive same as battery trying increase current keep current same inertia of current.

Self inductance L = N φ_B/I SI unit Henry H: 1 H = 1 Wb/A = 1 V·s/A. For long solenoid N turns length l area A L = μ0 N² A/l = μ0 n² A l where n=N/l turns per length. So L ∝ N².

Energy stored in inductor: Work done against self induced emf to build current I from 0 to I, energy stored as magnetic field U = ½ L I² Joules similar to capacitor U=½ C V² electric field. Energy density u = B²/(2μ0) magnetic field contains energy.

Inductor opposes change in current, in DC steady state acts as short wire zero resistance if ideal, in AC opposes AC frequency dependent reactance X_L=ωL.

Mutual induction phenomenon where changing current in one coil induces emf in nearby second coil because flux through second coil due to first coil changes flux linking.

Flux through coil 2 due to current in coil1 φ2 ∝ I1 so N2 φ2 = M I1 where M mutual inductance constant depends geometry N1 N2 A l μ0 separation orientation. Mutual induced emf in coil2 e2 = -d(N2 φ2)/dt = -M dI1/dt. Similarly e1=-M dI2/dt mutual same both ways M12=M21=M in linear medium.

Mutual inductance M = N2 φ2/I1 SI unit Henry H same as self inductance. For two coaxial solenoids long one inside other N1 N2 area A length l M=μ0 N1 N2 A/l=μ0 n1 n2 A l where n1=N1/l n2=N2/l. If coils far apart M small, close and wound on same iron core M large because μr large.

Flux linkage: φ2=M I1 flux through 2 due to 1, φ1=M I2 flux through 1 due to 2.

Transformer based on mutual induction: Primary coil AC current produces changing flux in core linking secondary coil induces emf in secondary energy transferred via changing magnetic flux through shared core no electrical connection. Efficiency high.

Mutual inductance depends on geometry N1 N2 A l μ0 separation orientation linear medium constant.

AC generator converts mechanical energy to electrical energy using electromagnetic induction. Principle rotating coil in uniform magnetic field produces changing flux induces alternating emf.

Construction: Rectangular coil N turns area A rotates with angular velocity ω in uniform magnetic field B between pole pieces. Slip rings and brushes collect current. Flux through coil φ=N B A cosθ where θ angle between normal to coil and B, θ=ωt if rotation uniform starting θ=0 when normal along B.

Induced emf e=-dφ/dt=-d/dt N B A cosωt=N B A ω sinωt=e0 sinωt where e0=N B A ω peak emf maximum amplitude. So emf sinusoidal alternating AC changes sign every half rotation. Frequency f=ω/2π equals rotation frequency.

If coil rotation uniform emf sinusoidal AC. Mechanical energy required to rotate coil against magnetic force due to induced current Lenz law opposing motion mechanical power converted to electrical power.

Slip rings rotate with coil brushes stationary transfer current to external circuit provide AC output to circuit.

DC generator similar but uses split ring commutator instead of slip rings to produce unidirectional DC pulsating.