Download = ∫ Physics 121 (Physics 2) Formulas, page 1 of 2 ∫= ∫

Physics 121 (Physics 2) Formulas, page 1 of 2
Area of circle = πr2 Circumference of circle = 2πr 1 meter = 1000 mm = 100 cm 1 kg = 1000 g
Surface area of sphere = 4πr2 Volume of sphere = (4/3)πr3 , 1 µC = 10-6 C 1 nC = 10-9 C
1/4πεo = ke = 9 x 109 N-m2/C2 , εo = 8.85 x 10-12 C2/N-m2, µo = 4π x 10-7 T-m/A
e = 1.60 x 10-19 C , me = 9.11 x 10-31 kg
F=
Point charges:
1 q1q2
r̂
4 πε 0 r 2
E=
1 Q
r̂
4 πε 0 r 2
where
r̂ is
a unit vector
ke ≡
1
= 9 x10 9
4 πε 0
n Superposition: contributions to the field or force from point Fnet on 1 = ∑ F1,i = F1,2 + F1,3 + F1,4 + .....
charges add as vectors at a point of interest
i= 2
Shell Theorem (spheres only): mimics point charge outside; inside E or F is zero
E = force per unit test charge at a point
F = qE
Fnet = ma
p 1
Dipole moment:
p = qd τdipole = pXE Udipole = -p.E
Eon dipole axis ≈ +
for
2πε
πε 0 z 3
For continuous charge distributions:
E=
∫
dist
ke
dq
.r̂
r2
(integrate over the distribution)
σ = surface charge density E cond sheet = σ/εo Enon-cond sheet = σ/2εo Eline = (1/2πεo)λ/r
dΦE = E . ndA = EAcos(φ)
ΦE = electric flux = qenc /εo = ∫ E . dA over a Gaussian surface
∆V = ∆U/q = - ∫ E cosθ ds = -
V – Vo = -E(x – xo)
Q = CV
∫
E . ds
∆Uel = q∆V
V = keQ/r
2
Electrostatic PE: Uel = Q /2C = CV /2
Cparallel = ΣCi 1/Cseries = Σ (1/Ci ) Cseries= C1C2 /( C1+C2 )
Dielectric constant: Cdie; = κ Cvac
Csphere = 4πεoR
q = ∫ idt = i∆t
R=V/i
dq = idt
i = dq/dt
R = ρL/A
V = iR
U = keQq/r
2
∆W nc = ∆Emech = ∆K +∆U
Cparallel plates= κεoA/d
i = ∫ J • d2 A = J∆A J = qnvdrift
ρ = ρ0 (1 + α(T – T0))
E = - dV/dx
J = σE σ = 1/ρ
Ohms Law: R independent of V
Rseries = ΣRi 1/Rparallel = Σ1/Ri Rpara = R1R2 /( R1 + R2) P = dUel/dt = iV
Presistor = i2R = V2/R
Junction rule: Σiin = Σiout
Loop rule: Σ ∆Vi = 0 around any closed circuit path. Follow currents.
RC Circuits: RC = time constant for circuit
charging: i(t) = (V/R)e-t/RC
Q(t) = CVcap(t) = CV∞ (1 - e-t/RC)
discharging: i(t) = (Qo/RC)e-t/RC
Q(t) = Qoe-t/RC
Fm =
τ = µxB U = -µ . B |µ|
µ| = NiA normal to loop
period = 2πm/(qB) ω = qB/m
f = ω /2π = 1/period
q vxB Fe = qE
Fm = i LxB
Cyclotron motion : r = mv/(qB)
Biot Savart: dB = (µo/4π)(i ds sinθ)/r2 where dB is in the direction of ds x r
µo = 4π x 10-7 T-m/A
Bwire = µoi / 2πr Barc = µoiφ /4πR Bcircle = µoi /2R Bsolenoid =µ0in F = µoi1i2 L /2πd (2 straight wires)
Amperes law:
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∫ B ds
= µ 0ienclosed
4/11/2012
for a closed “Amperian” loop
large z
Physics 121 (Physics 2) Formulas, page 2 of 2
Magnetic flux: dΦB = B.dA
Faraday’s Law:
ΦB =
∫
Eiind = - N dΦB/dt
Eind = BLv (slidewire)
Eself-induced = - L di/dt
B.dA
ΦB =BAcos(θ)
ΦB = 0 over every Gaussian surface
Lenz’s Law: induced flux, current, & emf oppose the change in ΦB
Eind = NABω sin(ωt) (rotating coil)
L = NΦB/i
Magnetic energy: UB = Li2/2
LR circuits: L/R= inductive time constant =τL
growth phase: VL(t) =
E e-Rt/L
Decay phase : VL(t) = -Rio e
LC circuit: Resonance at
LCR circuit: driven at
i(t) = Imaxsin(ωDt - φ)
ωres = 1 /
-Rt/L
i(t) = iinfinity(1 - e-Rt/L)
iinfinity = E/R
i(t) = io e-Rt/L
i0 = E/R
LC . No damping
ωd= 2πf
E(t) = Emaxsin(ωDt)
Em
Impedance: Z ≡ R 2 + (X L - X C ) 2
Reactances: XC = 1/ωdC
XL = ωdL
Voltage across inductance leads the current by 90o
Voltage across capacitance lags the current by 90o
Resonance occurs at ωd = ωres = 1 / LC
Phase angle:
tan(φ) = (XL – XC)/R
cos(φ) = R/Z
Irms = Imax/ 2
Vrms = Vmax/ 2 Irms = Vrms / Z
Pavg = IrmsVrmscos(φ) = I2rmsR
Transformer: Vs/Vp = Ns/Np = Ip/Is
Φ
VL-VC
Im
VR
Z
ωDt-Φ
Φ
X L- X C
R
Prefixes: n (nano) = 10-9 , µ (micro) = 10-6 , m (milli) = 10-3, M (Mega) = 10+6
Useful Integrals:
∫ dx / (a
2
∫x
n
dx = x n + 1 n + 1
+ x 2 ) = (1 / a ) tan −1 ( x / a )
∫ dx x + a = ln(x + a) ∫ e
∫ dx / (a
2
+ x 2 )3/2 = x / (a 2 a 2 + x 2 )
± αx
dx = ± e αx / α
∫ dx /
a 2 + x 2 = ln( x + a x + x 2 )
Physics 1: v = vo + at
x – xo = vo t + ½at2
v2 = vo2 + 2a(x – xo)
x – xo = ½(v + vo)t
ac = v2 /r
Fnet = ma = dp/dt
τnet = I α = dL/dt τ = r x F L = r x p
Vectors: Components: ax = a⋅cos(θ) ay = a⋅sin(θ) a = axi + ayj
| a | = sqrt[ ax2 + ay2 ] θ = tan-1(ay/ax) Addition: a + b = c implies cx = ax + bx, cy = ay + by
Dot product: a⋅b = a⋅b⋅cos(φ) = axbx + ayby + azbz unit vectors: i⋅i = j⋅j = k⋅k = 1; i⋅j = i⋅k = j⋅k = 0
Cross product: | a x b | = a⋅b⋅sin(φ); c = a x b = (ay⋅bz − az⋅by )⋅i + (az⋅bx − ax⋅bz )⋅j + (ax⋅by − ay⋅bx )⋅k
a x b = − b x a, a x a = 0 always; c = a x b is perpendicular to a-b plane; if a || b then | a x b | = 0
i x i = j x j = k x k = 0,
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ixj=k
4/11/2012
jxk=i kx i=j