Physics 1302
Equation Sheet
Some useful formulae:
Mechanics (from 1301):
P~
2
ac = vr (Centripetal acceleration)
F = m~a (Newton’s second law)
Ei + W = Ef (Work-energy theorem) E = K + U (Mechanical energy)
K = 21 mv 2 (Kinetic energy)
Electric Fields and Forces:
~ = k 2q r̂ (Electric field of a point charge)
E
F~12 = k rq21 q2 r̂12 (Coulomb force)
r
12
~
~
F = q E (Electric force on a charge q in external field)
~ (Dipole torque)
~ (Dipole energy)
p~ = qRd~ (Dipole Moment) ~τ = p~ × E
U = −~p · E
kr̂
~ =
dq (Electric field of a continuous charge distribution)
E
H
Rr2
~ · dA
~ = Qenc (Gauss’s Law)
~ · dA
~ (Electric flux)
ΦE = E
ΦE = E
S
S
ǫ0
Electric Potential:
Rf
~ · d~l (Electric Potential Difference)
∆V = Vf − Vi = − i E
R
kq
V = r (Potential of a point charge) V = k rdq (Potential of finite continuous distribution)
~ = −∇V
~ (Relationship between field and potential)
or E
Er = − ∂V
∂r
U = qV (Potential energy of a charge q in a potential V )
U = 21 QV (Self-energy of charged conU = 21 Σqi Vi (Energy of series of point charges)
ductor)
Capacitance:
2
C = Q/V
C = ǫ0 Ad (Parallel plate capacitance) U = 12 CV 2 = 21 QC (Energy of a capacitor)
ue = 21 ǫ0 E 2 (Energy density of electric field in vacuum)
Ceq = C1 + C2 + ... (Capacitors in parallel)
1/(Ceq ) = 1/C1 + 1/C2 + ... (Capacitors in
series)
E = Eκ0 (Field in dielectric placed in an external field E0 ) ǫ = κǫ0 (Permittivity of dielectric)
C = κC0 (Capacitance of a capacitor completely filled with dielectric)
Resistance
R and circuits:
∆Q
~ (Current definition where J = nevd , n = number density of charge carriers)
i = ∆t = J~ ·dA
R = ρ(L/A) (Resistance-resistivity relation)
ρ − ρ0 = ρ0 α(T − T0 ) (Resistivity-temperature relation)
ρ = nem2 τ (resistivity in metal, τ = mean time between collisions)
2
V = iR (Ohm’s Law) P = iV = i2 R = VR (Resistor power)
P = iE (Battery power)
Req = R1 + R2 + ... (Resistors in series)
1/Req = 1/R1 + 1/R2 + ... (Resistors in parallel)
Kirchhoff’s rules: (1) Sum of potentials/voltages around closed loop in circuit = 0.
(2) Net current flowing into node = 0.
RC circuits: Charging a capacitor through single resistor Q = Qf (1 − e−t/τ ) and i = i0 e−t/τ
Physics 1302 Eqn. Sheet
1
where τ = RC, i0 = E/R
Discharging a capacitor through single resistor Q = Q0 e−t/τ and i = i0 e−t/τ where i0 = V0 /R
Magnetism:
~
~ ×B
~
~ (B-force on moving charge/current)
F~ = q~v × B
F~ = iL
dF~ = id~l × B
~
~
~
µ
~ = N iA
~τ = ~µ × B
U = −~µ · B
(Magnetic dipole)
~
µ
id
l×r̂
0
~ =
dB
(Biot-Savart Law for B-field of current piece)
4π r 2
H
~ · d~l = µ0 ienc (Ampere’s Law)
B
C
Electromagnetic
Induction:
R
~ · dA
~ (Magnetic flux through a surface S)
ΦB = S B
H
~ · d~s and N = number of loops (Faraday’s Law)
E = −N dΦdtB where E = C E
N ΦB
di
E = −L dt where L = i (Inductor)
B2
uB = 2µ
(B-field energy density)
UB = 12 Li2 (Energy of inductor)
0
E
−t/τL
i = R (1 − e
) where τL = L/R (Rise of DC current in inductor)
−t/τL
i = i0 e
(Decay of DC current in inductor)
E2 = −M didt1 (Mutual inductance)
AC Circuits:
Irms = √12 Imax (for sinusoidal current, same for voltage)
1
(inductive and capacitive reactance)
XL = ωL, XC = ωC
For LRC series circuits: If E = Emax cos(ωd t) then i = Emax
cos(ωd t − φ)
Z
2
2
Pave = Irms
Rp= Irms
Z cos φ
C
where Z = R2 + (XL − Xc )2 , tan φ = XL −X
R
VS
NS
= iiPS = N
(Transformer)
VP
P
EM
Waves and Maxwell’s Equations:
R
H
~l = µ0 (ienc + id ) = µ0 ienc + µ0 ǫ0 d
~ · dA
~ (Ampere’s law with Maxwell fix)
~
E
B
·
d
dt S
C
~ ave = uave c = Erms Brms = E0 B0 (Average intensity of EM wave, where E = Bc)
I = |S|
µ0
2µ0
R
~ B
~
E×
~ · dA)
~
~
(Poynting vector, where energy flux [i.e., power] P = S
S=
µ0
p = Uc (Momentum of EM wave)
Pr = Ic (Radiation pressure, w/ 100% absorption, of EM wave)
I = I0 cos2 θ (Law of Malus for polarized light)
s
Magnetism in materials?:
constants:
g = 9.8 m/s2
e = 1.6 × 10−19 C
1
k = 4πǫ
= 9.0 × 109 N·m2 /C2
ǫ0 = 8.85 × 10−12 C2 /(N·m2 )
0
µ0 = 4π × 10−7 T·m/A (vacuum permeability)
c = √µ10 ǫ0 = 3.0 × 108 m/s (Speed of light in vacuum)
Physics 1302 Eqn. Sheet
2
√
2
If ax2 + bx + c = 0, then x = −b± 2ab −4ac .
~ ~
~ × B|
~ = AB sin θ
A
θ = Ax Bx + Ay By + Az Bz
|A
R · nB = AB1cosn+1
+ C forRn 6= −1 and ln |x| + C for n = −1
R x dx = n+1 x
cos θ dθ = sin θ + C and sin θ dθ = − cos θ + C
Physics 1302 Eqn. Sheet
3