Electromagnetism

Electromagnetism

1. Magnetic Field and Magnetic Force

   • Magnetic Flux
     • φB = BAcosθ
     • magnetic flux through any closed surface is always 0
       • B⃗ · d A⃗ = 0
     • E⃗ · d A⃗ = Q/E0
   • Magnet
     • North and south poles
     • Like poles repel, unlike poles attract
       • Either pole can attract metal
     • Spin order produces magnetism
   • Magnetic Field
     • Goes from north to south
     • enclosed
     • not lines of force
   • Magnetic Force
     • F = qvBsinθ F = q ⃗v × B⃗
       • Direction is determined by using right hand rule
     • does not do work
     • always towards the center
       • centripetal acceleration
     • particles have constant speed
       • v2 F = qvB = m v^2/r
         • r = mv/ qB
     • current carrying conductor
       • F = BIlsinθ F = I ⃗l × B⃗
         • right hand rule
   • Torque on current loop and electric motor
     • τ =F a+F a=BIba×2=BIab=BIA
     • τ = BIAsinθ
     • τ = NBIAsinθ = μBsinθ = μ⃗ × B⃗

2. Source of Magnetic Field

   • moving charge
     • dB⃗ = μ0 q⃗v × rˆ/ 4π r2
   • current carrying conductor
     • dB⃗ = μ0 Id⃗lrˆ /4π r2
     • center of current loops
       • dBx = dBcosθ
         • B = N μ0I /2R
   • Ampere's Law
     • magnetic field of closed path
       • B⃗ · d⃗l = μ0Iencl
     • magnetic field of straight wire
       • B = μ0I/2πr
     • two parallel conductors
       • F/l = μ0I1I2 / 2πd
       • currents in same direction, forces attract current in opposite direction, forces repel
     • Solenoid/ electromagnet
       • B=μ0nI=μ0(N/l)I

3. Electromagnetic Induction

   • Magnetic flux
     • φB = BAcosθ
     • magnetic flux through any closed surface is always 0
       • B⃗ · d A⃗ = 0
     • E⃗ · d A⃗ = Q /ε0
   • Faraday's Law
     • ε = −dφB/ dt
   • Len's Law
     • An induced current or emf always tends to oppose or cancel out the change that caused it.
   • Motional EMF
     • |ε|=Blv= (⃗v×B⃗)·d⃗l

4. Inductance

   • Self-inductance or Inductance: a changing current i in any circuit causes a self-induced emf.
     • ε = −LdI /dt
       • L=NdφB/dI =NφBI
     • Solenoid
       • L=NφB/I =NBAcosθ/I =Nμ0nIA =μ0N2A/l
     • toroidal solenoid
       • L=NφB/I =NBAcosθ/I =Nμ0NiA/2pir =μ0N2A/I
   • Mutual-inductance: when a changing current i1 in one circuit causes a changing flux in a second circuit, an emf ε2 is induced in the second circuit.
   • RL circuits
     • ε−IR−L ∆I/∆t=o
     • Current growth I = ε/R(1−e−t/τ)
     • Current decay I = (ε/R) e−t/τ
   • energy stored in magnetic field
     • U = 1/2 LI^2
     • U = 1/2C(∆V)^2

5. Alternating Current

   • Transformers can be used to step the voltage up and down with AC. High voltage and low current to minize i2r loss in cables.
   • voltage, current and phase φ
     • i = Icosωt v = V cos(ωt + φ)
   • Root mean square current
   • Reactance
     • φ = −90◦, voltage across a capacitor always lags the current by 90◦. Capacitive reactance
       • XC= 1/ 2πfC VC,rms = IrmsXC
     • φ = 90◦, voltage across a inductor always leads the current by 90◦. Inductive reactance
       • XL =2πfL VL,rms = IrmsXL
   • RLC circuit
     • Average power in an AC circuit Pav = IrmsVrmscosφ
     • Resonance frequency in a series RLC circuit f0 =1/ √ 2π LC ω0 = 2πf0
       • Current gets maximum and impedance becomes minimum Z = R at the resonance frequency.
   • The transformer
     • ε1 =−N1dφB/dt ,ε2 =−N2dφB/dt
       • V2/N2 = V1/N1
     • If N2 > N1, step up tranformer. If N2 < N1, step down tranformer. Ideal tranformer I1VI = I2V2

6. Electromagnetic waves

   • E (x, t) = ˆjEmaxcos(kx − ωt) B⃗ (x, t) = kˆBmaxcos(kx − ωt)
   • Transverse waves: E⃗ and E⃗ are perpendicualr to the direction of propagation of waves.
     • E/B = Emax/Bmax = 1/ε0μ0
   • EM waves carry energy. Intensity I, average power per area.
     • I= EmaxBmax / 2μ0 C = fλ