FORMULA SHEET FOR PHYS. 215 FINAL EXAM
Definition of electric current:
Sum of two vectors F1 and F2 , with magnitude F1 and F2 , and components (F1x = F1 cos θ1 , F1y =
F1 sin θ1 ) and (F2x = F2 cos θ2 , F2y = −F2 sin θ2 ). The vector F ≡ F1 + F2 has magnitude and direction:
q
F = Fx2 + Fy2
tan θ =
Fx = F1x + F2x = F1 cos θ1 + F2 cos θ2 ,
Coulomb’s Law:
Fy
,
Fx
I=
Ohm’s law:
V = IR ,
1
k=
.
4π0
R=ρ
ρ(T ) = ρ0 [1 + α(T − T0 )] ,
Electric field for a single charge Q:
E=
P Ea
,
q
F
q
For alternating currents:
√
I → Irms = I0 / 2 ,
1 Q
.
4π0 r2
√
V → Vrms = V0 / 2 ,
V2
.
R
I0 = peak current, V0 = peak voltage.
Microscopic view of electric current in a wire with cross area A:
P Ea = electric potential energy at point a.
I = neAvd ,
n = number of electrons per unit volume, vd = drift velocity of the electrons.
First Kirchoff law [positive (negative) currents entering (leaving) the junction.]:
∆P E
∆W
∆V =
=
,
q
q
∆V ≡ Vb − Va ,
∆P E ≡ P Eb − P Ea ,
σ
E=
,
0
I1 + I2 + . . . + In − In+1 − In+2 . . . − In+m = 0 ,
etc . . .
Potential difference for a parallel plate capacitor:
d = distance between the plates, σ = the charge per unit area.
Second Kirchoff law:
V − I 1 R1 − I 2 R2 . . . − I n Rn = 0 ,
Resistors in series:
R = R 1 + R2 + R3 + . . .
Electric potential for a single charge Q:
Resistors in parallel:
V =
1 Q
.
4π0 r
R=
1 Ql cos θ
.
V =
4π0 r2
V1 =
Q
V
Capacitance for a parallel plate capacitor with plate area A and plate distance d:
K = dielectric constant, = K0 = permittivity of the dielectric.
R1
V,
R1 + R 2 + . . .
I1 =
1/R1
I,
1/R1 + 1/R2 + 1/R3 + . . .
U=
1 Q2
1
1
QV = CV 2 =
.
2
2
2 C
1
R2
V,
R1 + R 2 + . . .
I2 =
1/R2
I,
1/R1 + 1/R2 + 1/R3 + . . .
...
Emf and terminal voltage:
∆V = E − Ir ,
r = internal resistance of the battery, E = EMF.
Capacitors in series:
Energy stored in a capacitor:
V2 =
Partition of currents in junctions:
Definition of capacitance:
C=
1
.
1/R1 + 1/R2 + 1/R3 + . . .
Potential drop across resistors in series:
Electric potential at large distance r of a dipole with charges ±Q and axis length l:
A
A
C = K0 ≡ ,
d
d
α = temperature coefficient of resistivity.
P = IV = I 2 R =
Potential difference:
∆V = Ed ,
ρ = resistivity.
Electric power:
Definition of electric potential at point a (q = test charge):
Va =
L
,
A
Variation of resistivity with temperature T :
Definition of electric field (q = test charge):
E=
R = resistance.
Resistance for a wire with cross area A and length L:
Fy = F1y + F2y = F1 sin θ1 − F2 sin θ2 .
Q1 Q2
F =k 2 ,
r
∆Q
.
∆t
C=
1
.
1/C1 + 1/C2 + 1/C3 + . . .
2
...
Capacitors in parallel:
Velocity of light in a medium:
1
v=√ .
µ
C = C 1 + C2 + C3 + . . .
Force F on an electric current I in a magnetic field B:
Relation between frequency ν and wavelength λ:
F = IlB sin θ .
λ=
Force F on an electric charge q with speed v in a magnetic field B:
Mirror equation:
F = qvB sin θ .
Magnetic field due to a straight wire:
1
1
1
+
= ,
do
di
f
µ0 I
B=
.
2π r
do = obj. distance, di = image distance, f =
F
µ0 I 1 I 2
=
.
l
2π L
di
hi
=− ,
ho
do
m=
ho = object height, hi = image height.
Index of refraction:
n=
Ampère law:
c
.
v
Snell’s law:
B|| ∆l = µ0 I
Magnetic field inside a solenoid with n loops per unit length:
ni sin θi = nr sin θr ,
θi (ni ) and θr (nr ) = incident and refraction angles (indices of refraction).
Total internal reflection:
B = µ0 nI .
sin θc =
Magnetic flux:
n2
,
n1
n2 < n 1 .
Power of a (thin) lens:
ΦB = BA cos θ .
Faradays’ law:
E =−
P =
Thin lens equation:
∆ΦB
.
∆t
Emf produced by an electric generator:
E = N BA ω sin ωt ,
Transformers:
R
= focal length of the mirror w/ radius R.
2
Lateral magnification of a mirror:
Force between two straight wires of length l and distance L:
X
c
.
ν
1
1
1
+
= .
do
di
f
Lateral magnification of a thin lens:
N = number of loops, ω = angular frequency of the rotor.
m=
VS
NS
=
,
VP
NP
P = primary, S = secondary.
hi
di
=− .
ho
do
Wavelength of light in a medium with index of refraction n:
λn =
Inductance between two coils:
E1 = −M
1
.
f
∆I2
,
∆t
E2 = −M
Self-inductance:
E = −L
∆I
.
∆t
∆I1
.
∆t
Double slit (or diffraction grating) interference [d=slit distance, m=fringe order]:
1
λ (dark fringes)
d sin θ = mλ (bright fringes) d sin θ = m +
2
Single slit interference [D=slit width]:
D sin θ = mλ
Energy stored in a magnetic field:
1 B2
1
.
U = LI 2 =
2
2 µ0
Velocity of light in vacuum:
c= √
1
.
0 µ0
3
λ
.
n
(bright fringes)
Interference of a thin film with thickness t [θ=incidence angle]:
1
λ cos θ
2t = mλ cos θ
or
2t = m +
2
depending on the index of refraction of the two materials at the reflection boundary.
4