Exam II Equation Review
Ch. 27 Current and Resistance
• i=
dq
dt
• Current density: i =
Z
~
J~ · dA
• Current and Drift speed: J~ = (nq)v~d and i = nqvd A
• Resistance: R =
V
i
; Resistivity: ρ =
E
J
L
• R = ρA
Also P = i2 R = V 2 /R
• Power in electric circuits: P = iV
Ch. 28 DC Circuits
• EMF = ε = open circuit voltage of battery.
• ∆V = ε − Ir terminal voltage across battery with internal resistance r.
• Resistors in series: Req = R1 + R2 + ...
• Resistors in parallel: 1/Req = 1/R1 + 1/R2 + ...
• Kirchhoff’s Rules.
– Junction Rule: Σjunction I = 0
– Loop Rule: Σclosedloop ∆V = 0
• RC Circuits
–
–
–
–
Charging current: I(t) = I0 e−t/RC
Charging charge: q(t) = Qmax (1 − e−t/RC ), where Qmax = Cε
Discharging current: I(t) = I0 e−t/RC
RC time constant, τ = RC, is time to change by 63%.
Ch. 29 Magnetic Fields
~
• F~B = q~v × B
~ + ~v × B)
~
• Electric and Magnetic combined: F~B = q(E
• Circulating charged particle: qvB =
• Circulating charged particle: ω =
mv2
r
qB
m
• Helical paths: v⊥ = v sin φ
~ ×B
~
• Force on current carrying wire: F~B = iL
• Know mass spectrometer, velocity selector, cyclotron
Ch. 30 Magnetic Fields Due to Currents
• Biot-Savart Law
~ =
dB
µ0 id~s × ~r
4π r 3
1
OR
~ =
dB
µ0 id~s × r̂
4π r 2
~
• Right-hand rule for cross products (like d~s × ~r and ~v × B)
~ around wires.
• Right-hand rule for finding direction of B
• → thumb points in direction of current
• B-field for long straight wire: B =
• B-field at center of circular arc: B
µ0 i
2πR
0 iφ
= µ4πR
~ ×B
~a
• Force between two parallel currents: F~ba = ib L
µ0 ia
,
2πd
µ0 ia ib L
(direction
2πd
• → since Ba is
• → Fba =
determined by RHR or remember ”opposites attract”.)
• Ampere’s Law
I
~ · d~s = µ0 ienc
B
• Solenoids and Toroids
• B-field in ideal solenoid: B = µ0 in
• B-field in toroid: B =
µ0 iN
2πr
• A current-carrying coil as a magnetic dipole: ~µ = NiA
H
~ · dA
~=0
• Gauss’s law of magnetism: B
Ch. 31 Faraday’s Law of Induction
• Faraday’s Law: ε = −dΦB /dt (EMF around loop of wire.)
• Faraday’s Law, N loops: ε = −NdΦB /dt
• Motional EMF: ∆V = Blv between top and bottom of conducting bar moving at constant
v through uniform ⊥ B
H
~ · d~s = −dΦB /dt
• Faraday’s Law, general, no wire needed: E
• Generators: EMF generated by loop rotating in uniform B: ε = NABω sin ωt
• Motors have the same mechanical structure as a generator, but are run “in reverse” (i.e.,
current pushed through a loop causing the loop to turn).
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