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PHY2049 Formulas: Sheet 1
Chapter 21 – 23 (Electric forces and fields, Gauss’ law)
Coulomb’s Law 2 2 0
kQq Qq
r πε r
F = r ≡ r (point charge) r ˆ = unit vector from Q to q.
Electric field F = q E (general) 2 2 0
kQ Q
r πε r
E = r ≡ r (single point charge Q )
2 2
ˆ (^) or ˆ i i i (^) i
kq kdq
r r
= (^) ∑ = ∫ E r E r (point charges or continuous)
Electric dipole U = − p E ⋅ = − pE cos θ τ= p × E τ= pE sinθ
G
Gauss’ law
encl
0
E (^) i i i S
Q
d
Φ = (^) ∑ ⋅ = ⋅ = ∫
E A E A
v Φ (^) E ="electric flux"
Chapter 24 – 25 (Electric potential, capacitors)
Work 1 2 1 2 2 2
f f i f i i
W = ⋅ d = K − K = mv − mv ∫ F s
Conservative force
f f i i i f f i
U − U = − ⋅ d → U + K = U + K ∫ F s (energy conservation)
Electric potential
0
(general) (point charge ) 4
U kQ Q V V Q
q r πε r
Potential difference Δ V ≡ V f − V i = − E ⋅ ( x f − x i )( E constant)
f f i i
V − V = − ⋅ d ∫ E s ( E variable)
E field from V (^) x y z
V V V
E E E
x y z
Capacitors
0 A
C
d
= (flat plates) ( )
ln /
L
C
b a
= (cyl shell) (^4 )
ab C b a
(sph shell)
Capacitors (cont) (^) eq 1 2
eq 1 2
(parallel) (series)
E
q CV E C C C C C C C C
Energy
2 1 2 1 2 2 (energy)^20 (energy density) 2
E E
q U CV u E C
Chapter 26 – 27 (Electric current, circuits)
Current (basic def) (using current density) S
dq i i JA i d dt
∫
J A
Current density 1 (using conductivity or resistivity ) ρ
J = σ E = E σ ρ
Drift velocity
2 / (resistivity) e d e e
J = en v ρ= m e n τ
Resistance eq 1 2 eq 1 2
(series) (parallel)
L
V iR R R R R A R R R
Temp dependence ( )
0 0 0
ρ − ρ = ρ α T − T
RC circuits ( )
/ / max max 1 RC^ (charging) RC (discharging)
t t RC RC q q e q q e
τ τ
− − = = − =
Circuits (1) Current entering junction = current leaving junction (2) 0 (over loop) i i
∑ V^ =
Power in circuit
2 P = iV (general power eqn) P = i R (power lost in resistor)
Chapter 28 – 29 (Magnetic fields)
Magnetic force F = q v × B F = qvB sin φ (charge) F = i L × B F = iLB sin φ (current)
Magnetic dipole μ = NiA (current loop)
τ = μ × B τ = μ B sin θ (torque) U = −μ ⋅ B = −μ B cos θ (potential energy)
G G G
Generating B field 0 0 2 2
ˆ (^) sin (Biot-Savart law) 4 4
id ids d dB r r
×
s r B
0 0 enc (Ampere's law)^ (long wire) 2
i d i B r
B s
v
0 0 (circular loop) (partial loop) 0 (solenoid) 2 4
i i B B B ni R R
Force between currents 0
2
a b ab
i i L F d
B field of dipole ( )
0 3 2
z z
B =
G
(very far from dipole)
Thermal energy 3 2 B
K = k T per molecule
EM waves sin (^) ( ) sin (^) ( ) 2 / 2 / m m E = E kx − ω t B = B kx − ω t k = π λ ω = π f c = λ f =ω k
1 1 0 0 max rms 2 2
E = Bc c = 1/ μ ε Em = E B = Bm
Intensity and power
2 2 ave rms rms 0 0 0
1 1 c I S E B
μ c μ μ
S = E × B = = =
2
(abs), (refl) 4
r
P I I
I p π r c^ c
Polarization
(^2 )
I = I 0 cos θ I = 2 I unpolarized
Refraction 2 2 1 1 2 2 1 1
sin sin sin (^) c tan B
n n n n n n
Chapter 34 (Images)
Lens equations (^) ( ) 1 2 1 2
(mirror) 1 (lensmaker's eqn)
i m f R n p i f p f r r
Magnifier
25cm 25cm m (image at ), 1 (maximum) f f
θ
Telescope, microscope
obj
eye obj eye
(telescope) (microscope)
f m M f f f
θ
Spherical lens
n 1 (^) n 2 (^) n 2 (^) n 1
p i r
Chapter 35 – 36 (Interference and diffraction)
Interference / 2 / path length n n
λ = λ n Δφ = π Δ L λ Δ L =
Thin film ( )
1 1
1
2 d = m λ n (constructive) = m + 2 λ n (destructive) ( n 0 < n 1 < n 2 )
Diffraction d sin θ = m λ(max, 2 slit or grating) a sin θ = m λ(min, single slit)
Δ θ = 1.22 λ/ D (Rayleigh criterion) Δθ ≈λ/ d (interferometer resolution)
Intensity diffraction
2 2 sin sin 2 sin sin m (1 slit)^ m cos^ (2 slit)
a d I I I I
Diffraction grating
2 2 sin sin ( slits) (half-width of maxima) sin cos
m hw
N
I I N
N Nd
⎛ ⎞ ⎛^ ⎞
avg (dispersion) (resolving power) cos
m D R Nm d
θ^ λ
