Download INTRODUCTION TO PHYSICS II FORMULA

PHYSICS 2000 : INTRODUCTION TO PHYSICS II
FORMULA SHEET, Spring 2014
Coulomb’s Law:
|F~ | =
k|q1 q2 |
r2
, k=
1
4π0
~ =
Electric field and force: E
= 9.00 × 109 N · m2 /C2
~ = k q2 r̂
Electric field due to
a point charge: E
a group of point charges:
r
Proton charge: e = 1.60 × 10−19 C
~ = R dE
~ = k R dq2 r̂
a continuous distribution of charge: E
r
~ = k Pi
E
~
F
q0
qi
r̂
ri2 i
Electric dipole moment p~: |~p| = |q|d , two charges q and −q separated by a distance d.
~ the torque on an electric dipole is:
~ ,
In uniform E,
~τ = p~ × E
~ .
and the potential energy of a dipole is:
U = −~p · E
2kp
θ
Field due to the dipole, on its axis:
Ez = z 3 ,
Potential due to dipole:
V = kp rcos
2
Electric flux through a
~
plane surface in uniform E:
Gauss’ Law:
ΦE =
~ ·A
~
ΦE = E⊥ A = E
H
~ · dA
~ = Qencl /0 ,
E
~
curved surface in any E:
R
~ · dA
~
E
net electric flux ΦE through any closed surface
equals the net charge enclosed Qencl divided by 0 .
1
(0 = 4πk
= 8.854 × 10−12 C2 /N · m2 )
~ = k|Qencl |/r2 .
|E|
~ = 2k|λencl |/r .
|E|
~ = |σleft − σright |/20 .
|E|
For any spherically symmetric charge distribution,
For any cylindrically symmetric charge distribution,
For any plane-symmetric charge distribution,
Change in potential energy:
∆U = Ub − Ua = −Wa→b = −
Potential difference:
~
in a uniform E:
∆V = ∆U /q = −
~ · ~` ,
∆V = −E
Rb
a
~ · d~`
E
Rb
a
R
~ · d~`
F~ · d~` = −q ab E
units: 1 volt = 1 V = 1 J/C
~` = displacement from a to b.
V = k qr .
Setting V = 0 at ∞, potential due to a point charge q a distance r from the charge:
Potential energy of a pair of point charges q1 , q2 :
ΦE =
U12 = k qr112q2 ,
r12 = the distance of separation.
Electric potential due to a continuous charge distribution:
V = k
∂V
∂V
∂V
Electric field from electric potential: Ex = − ∂x , Ey = − ∂y , Ez = − ∂z .
R dq
r
.
~ = 0 everywhere inside.
Conductor in electrostatic equilibrium: 1. E
2. excess charge resides on surface.
~
~
3. E just outside is perpendicular to the surface, |E| = σ/0 , where σ is the surface charge density.
4. surface is equipotential, & the potential is the same everywhere throughout the conductor.
The capacitance of any capacitor:
parallel plate capacitor:
isolated charged sphere:
C = |Q|/|Vab | ,
C = 0 A/d ;
C = 4π0 R ;
units: 1 farad = 1 F = 1 C/V.
A= plate area, d= plate separation.
R= radius.
Equivalent capacitance Ceq
capacitors in parallel: Ceq = C1 + C2 + C3 + . . .
Potential energy stored in capacitor: U =
Q2
2C
capacitors in series:
1
Ceq
=
1
C1
+
1
C2
+
1
C3
+ ...
~ uE := U/Vol. = 0 E 2 /2 .
= 12 QV = 12 CV 2 , energy density of E:
C0 = capacitance without dielectric inserted between conductors, C= capacitance with dielectric
inserted between conductors, C = KC0 , where K= the dielectric constant.
E.g., parallel-plate capacitor: C = K0 A/d = A/d , := K0 is the permittivity of the dielectric;
energy density in dielectric: u := K0 E 2 /2 = E 2 /2 ;
H
~ · dA
~ = Qencl−free /0 ,
Gauss’ Law: K E
where Qencl−free is the free charge (not bound charge) enclosed by the Gaussian surface.
R
~ .
,
units: 1 A = 1 C/s . J~ current density, I = J~ · dA
Electric current: I = dQ
dt
Conductor, cross-sectional area A, charge carriers of charge q, density n, drift velocity ~vd , the current is
I = nqvd A,
so that the current density is
J~ = nq~vd .
~ , where ρ is the resistivity of the material.
Ohm’s Law:
I = V /R and J~ = E/ρ
The resistance R of a conductor is
R = ρ`/A .
Unit of R: 1 ohm = 1 Ω.
Temperature dependence: ρ(T ) = ρ0 [1 + α(T − T0 )] , where α is the temperature coefficient of resistivity
of the material.
The power, or rate at which energy is absorbed by an electrical device, is P = IVab .
Emf E = dW
dq
Equivalent resistance Req
resistors in series: Req = R1 + R2 + R3 + . . .
resistors in parallel: R1eq = R11 +
1
R2
+
1
R3
+ ...
P
Junction Rule: I = 0 ,
sum of currents entering a node = sum of currents leaving that node.
P
Loop Rule: V = 0 ,
sum of potential differences across each element around a closed-loop circuit = 0.
RC Circuits:
capacitor charging up: Q(t) = Qf [1 − e−t/RC ], I(t) = I0 e−t/RC ;
capacitor discharging: Q(t) = Q0 e−t/RC , I(t) = (Q0 /RC)e−t/RC .
~
Magnetic force on a particle of charge q moving with velocity ~v in a magnetic field B:
~ 1 tesla = 1 T = 1 N/(A·m) .
Units of magnetic field B:
~ + q~v × B
~ .
When both magnetic and electric fields are present, F~ = q E
~ .
F~ = q~v × B
~ F~ = I ~`0 × B,
~
Force on a conductor carrying current I in a uniform magnetic field B:
~`0 starts where the current enters the conductor, and ends where it exits.
R
~ .
In a nonuniform magnetic field, F~ = I d~` × B
~ where A
~ is a vector whose
The magnetic moment µ
~ of a loop of N turns carrying a current I is µ
~ = N I A,
magnitude equals the area of the loop, and whose direction is perpendicular to the loop.
~ experiences a torque ~τ = µ
~ ,
A current loop (or any magnetic moment) in a uniform magnetic field B
~ ×B
~
and its potential energy is U = −~µ · B.
~ = µ0 q~v×r̂
Magnetic field of a moving charge: B
.
4π r2
~
Biot-Savart law: Magnetic field dB due to a current element Id~` carrying a steady current I is:
~ = µ0 Id~`×r̂
, where µ0 = 4π × 10−7 T·m/A and r is the distance from the element to the field.
dB
4π
r2
~ = R dB
~ = µ0 R Id~`×r̂
The total magnetic field due to the entire conductor is B
.
4π
r2
µ0 I
Magnetic field a distance a from a long, straight wire carrying a current I is
B = 2πa
.
The force per unit length between two parallel wires a distance a apart, carrying currents I1 and I2 , is
I1 I2
F
= µ02πa
; the force is attractive (repulsive) for currents in the same (opposite) directions.
`
H
~ · d~` = µ0 Iencl . That is, the line integral of B
~ · d~` around any
Ampere’s Law states
B
closed path (or Amperian loop) equals µ0 times Iencl , the total steady current encircled by that path.
From Ampere’s law, the field inside an ideal solenoid is B = µ0 N` I = µ0 nI .
~ = µ0 |Iencl |/(2πr) .
For any cylindrically symmetric current distribution,
|B|
~ = µ0 |(I/`)left − (I/`)right |/2 .
For any plane-symmetric charge distribution,
|B|
~ varies with time),
The generalized form of Ampere’s Law (generalized to situations in which E
or the Ampere-Maxwell Law:
H
~ · d~` = µ0 (IC + ID )encl = µ0 Iencl + µ0 0 dΦE ,
B
dt
where ΦE is the electric field flux through any surface bounding the closed path,
ID := 0 dΦdtE is the displacement current, and Ic = I is the conduction current.
R
~ · dA
~ .
Magnetic flux through a surface: ΦB = B
H
~ · dA
~ = 0,
Gauss’ Law of magnetism: B
the net magnetic flux through any closed surface is zero (there are no magnetic monopoles).
H
~ · d~` = − dΦB .
Faraday’s Law of Induction: emf E = E
dt
That is, the emf induced around a closed path (or loop) has magnitude equal to
the time rate of change of the magnetic flux through the loop.
For N turns, the induced emf is N times as big.
Lenz’s Law: the emf induced around a loop has the direction that creates, or would create, a
magnetic field that opposes the change in ΦB that produced them.
~
Motional emf: when a straight conducting bar of length vector ~` moves through a magnetic field B
R
~ · ~` . In general, E = (~v × B)
~ · d~`.
with a velocity ~v , the induced emf is E = (~v × B)
induced emf:
E2 = −M dIdt1
mutual inductance: M = N2IΦ1B2
emf of an inductor: emf = −L dI
, self-inductance L = N ΦIB (SI unit: 1 henry=1 H);
dt
ideal solenoid: L = µ0 N 2 A/` .
Energy stored in an inductor: U = LI 2 /2 , energy density of a magnetic field: uB = U/Vol. = B 2 /2µ0 .
CALCULUS
R n
R −1
d
xn+1
m
m−1
x
=
m
x
,
x
dx
=
(n
=
6
1)
,
x dx = ln |x|
dx
n+1
R
d
d
cos
x
=
−
sin
x
,
sin
x
=
cos
x
,
cos
x
dx
=
sin x ,
dx
R x dx
d x
x
e = e = e dx
dx
) df (x)
d
chain rule: dx
g f (x) = dg(f
df
dx
R
sin x dx = − cos x