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PHY2054 Fall 2015
PHY2054 Exam 2 Formula Sheet
Vectors
r
r
r r r r
a = a x xˆ + a y yˆ + az zˆ b = bx xˆ + by yˆ + bz zˆ Cross Product Magbitude: a × b = a b sin θ ab
r r r
Cross Product Vector: c = a × b = ( a y bz − a z by ) xˆ − ( a x bz − a z bx ) yˆ + ( a x by − a y bx ) zˆ
Electromagnetic Force
r
r
r
r
r r
Electrmagnetic Force (vector): FEM = FE + FB = qE + qv × B
r
r
FE = qE
r
r r
FB = qv × B
(r = distance between charge Q and charge q, v = velocity of charge q, V = velocity of charge Q)
r
Qq
FE = k 2 rˆ (units = N)
r
k = 1/(4πε0) ≈ 8.99x109 N·m2/C2
kB = k/c2 = µ0/(4π) ≈ 10-7 Tm/A
r
Qq r r
FB = k 2 2 v × V × rˆ (units = N)
cr
ε0 ≈ 8.85x10-12 C2/(N·m2)
µ0 ≈ 4π × 10-7 Tm/A
c ≈ 3 × 108 m/s (speed of light)
r
Q
E = k 2 rˆ (units = N/C = V/m)
r
r
Q r
Magnetic Field (due to Q): B = k B 2 V × rˆ (units = N/(C·m/s) = T)
r
r
I r
Magnetic Field (due to current I): B = k B 2 l × rˆ (units = N/(C·m/s) = T)
r
2
Energy Density (Electric & Magnetic Field): u E = 12 ε 0 E
u B = 2 µ1 0 B 2 (units = J/m3)
r
r r
Magnetic Force (on a long straight wire carrying current I): FB = IL × B (units = N)
r
r
r
Magnetic Dipole Moment (N loops, current I, area A): µ B = NIA (units = A·m2) A = Anˆ
r
r r
Magnetic Torque on a Magnetic Dipole: τ = µ B × B (units = N·m)
r r
Ampere’s Law: ∫ B ⋅ dl = ∑ B|| ∆l = µ0 I enclosed (around a closed loop)
Electric Field (due to Q):
C
C
Magnetic Field (Examples)
r
Infinite Straight Wire Carrying Current I: B = 2k B I / rperp (units = T)
r
Center of a Circular Loop Carrying Current I: B = 2πk B I / R (units = T)
r
Infinite Solenoid (current I, n loops per unit length): B = µ 0 nI (units = T)
Electromagnetic Induction, RL Circuits, and LC Circuits
r r
Magnetic Flux (uniform B, surface A): Φ B = B ⋅ A = BA cosθ = B perp A units = Tm2 = Wb
Faraday’s Law of Induction:
ε =−
∆Φ B
(ε = induced EMF, units = V)
∆t
Inductor (inductance L units = H): ∆VL = − L
∆I
(potential difference) U L = 12 LI 2 (stored energy)
∆t
RL Circuits (time constant): τL = L/R (units = H/Ω = s)
RL Circuits (EMF ε, Resistor R, Inductor L, swirch closed at t = 0):
I (t ) = ε (1 − e −t / τ L ) / R
ω = 1 / LC
I (t ) = I 0 cos(ωt + ϕ )
Oscillating LC Circuit (no resistance): U tot = 12 Q 2 / C + 12 LI 2 (stored energy)
Oscillating LC Circuit (no resistance): Q (t ) = Q0 sin(ωt + ϕ )
Oscillating LC Circuit (no resistance): f = ω/ 2π
π (frequency of oscillations in Hz)
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