Constants Chapter 22
19
Magnitude of electron charge e = 1.60 ⇥ 10 C q1 q2
Coulomb’s Law F~12
E
= k 2 r̂12
r12
Coulomb’s constant k = 8.99 ⇥ 109 N.m2 C 2
~r2 ~r1
Directional unit vector r̂12 =
1 r12
Permittivity of free space ✏0 =
4⇡k
12
✏0 = 8.85 ⇥ 10 C2 N 1
m 2
Permeability of free space µ0 = 4⇡ ⇥ 10 7
H.m 1 Chapter 23
31
Mass of electron me = 9.11 ⇥ 10 kg
Dipole moment (electric) p~ ⌘ qp~rp
Speed of light c0 = 3.0 ⇥ 10 m/s 8 P ~
Torque on electric dipole ~⌧ = p~ ⇥ E
~E
Electric field ~ ⌘ Ft
E
Mathematics qt
p
b± b2 4ac Induced dipole moment ~
p~ind = ↵E
2
Quadratic ax + bx + c = 0 x=
2a
Superposition of electric fields ~ =E
E ~1 + E
~2 + · · ·
~ =A
Adding vectors C ~+B
~ C x = Ax + B x
q
Uniform linear charge density ⌘
C y = Ay + B y `
q
~ = Ax î + Ay ĵ + Az k̂ Uniform surface charge density ⌘
Vector components A a
q
Uniform volume charge density ⇢⌘
✓ counterclockwise from x-axis Ax = A cos ✓, Ay = A sin ✓ V
~·B
~ = AB cos ✓ Electric field due to:
Scaler(Dot) product A
A point charge ~ s = k qs r̂sP
E
~·B
A ~ = Ax B x + Ay B y 2
8rsP
>
>
Vector(Cross) product ~ ⇥ B|
|A ~ = AB sin ✓ >
< 2kp if on y-axis
y3 ,
Dipole (aligned with y-axis, Ey ⇡
~⇥B
~ = ~ ⇥A
~ >
>
A B >
: kp
far from dipole) |x3 | , if on x-axis
Area sphere 4⇡r2
2k
An infinite line of charge Ex =
4 3 x
Volume sphere 3 ⇡r
s qz
A charged ring (on the axis) Ez = k
⌃ni=1 (xi xave )
2 (z 2 + R2 )3/2
Standard deviation =
n 1 1 1
A charged disk (on the axis) Ez = 2k⇡ z
|z| (z + R2 )1/2
2
Equations from 121 An infinite plane Ez = 2k⇡
8
>
>
>
<k rq2 , if r > R
Constant acceleration (x dir.) xf = xi + vx,i t + 12 ax t2
A thin spherical shell E=
>
>
vx,f = vx,i + ax t >
:0, if r < R
2 2
vx,f = vx,i + 2ax x
x = 12 (vx,i + vx,f ) t
Chapter 24
Kinetic energy K = 12 mv 2
P~ Z
F Electric flux = ~ · dA
E ~
Equation of motion ~a = E
m I
Interaction pair F12 = F~21
~ Gauss’s law E = E ~ = qenc
~ · dA
✏0
1
,Chapter 25 Chapter 14
q1 q2 1
Electric potential energy UE = `proper
4⇡✏0 r12 Z Length contraction `v =
B
Electrostatic work Wq (A ! B) = q ~ · d~`
E 1
A Lorentz factor ⌘q
v2
Potential (0 at 1) due to: 1 c20
1 X qn Time dilation tv = tproper
Point charges Vp =
4⇡✏0 rnP
1 q Lorenta trans frumation
A charged ring (on the axis) V =
tuew =
Z (t -
最 x ]
4⇡✏0 (z + R2 )1/2
2
⇣p ⌘ xew =
z ( x rt }
-
A charged disk (on the axis) V = z 2 + R2 |z|
2✏0 Chapter 28
1 q
A charged sphere (r > R) V =
4⇡✏0 r I
~ @V @V @V ~ · d~` = µ0 Ienc
Electric field (from potential) E= î ĵ k̂ Ampere’s law B
@x @y @z
~
Potential di↵erence VAB ⌘
Wq (A ! B)
Biot-Savart law ~ s = µ0 Id` ⇥ r̂sP
dB
4⇡ 2
rsP
q
I Z
Potential di↵erence closed path ~ · d~` = 0
E ~ =
B dB~s
current path
µ0 I
Magnetic field due to long wire B=
Chapter 26 2⇡r
Magnetic field due to solenoid B = µ0 nI
q
Capacitance C⌘ µ0 N I
Vcap Magnetic field due to toroid B=
✏0 A 2⇡r
Parallel plate capacitor C=
d
2⇡✏0 l
Coaxial cylindrical capacitor C=
ln(R2 /R1 )
V0
Dielectric constant ⌘ Chapter 31
Vd
Wnonelectrostatic
Emf E⌘
q J
2 Conductivity ⌘
Capacitor potential energy U E
= 1q
= 1 2 1 E
2 C 2 CVcap = 2 qVcap
ne2 ⌧
1 Conductivity metal =
Energy density uE = ✏0 E 2 me
2 l
Resistance R=
A
Chapter 27 |I|
Current Density J⌘
A
dq ~
eE
Current I⌘ Drift velocity ~vd = ⌧
dt me
⇣ ⌘
Electromagnetic force F~PEB = q E~ + ~v ⇥ B
~
Equiv. Resistance (series) Req = R1 + R2 + R3 + · · ·
I
Gauss’s law for magnetism = ~ · dA
B ~=0 1 1 1 1
B (parallel) = + + + ···
Req R1 R2 R3
B
~ Fw, max
Magnetic field (wire ? B) B⌘ Junction rule Iin = Iout
|I|`
Z
~ · dA
~ X X
Magnetic flux B = B Loop rule E+ V =0
Magnetic force F~wB = I ~` ⇥ B
~ = q~v ⇥ B
~ V
Ohm’s law I=
R
mv V
Particle in magnetic field R= Resistance R⌘
|q|B I
V2
Hall probe voltage across w V = vwB Power dissipated by a resistor P = V I = I 2R =
R
2
, Chapter 29 1
Resonant angular frequency !0 = p
LC
2π
=
f
1 B2 Xmax
Energy density uB ⌘ Rms Xrms = p
2 µ0 2
1
Inductor potential energy U B = 12 LI 2 Average power delivered Pav = Emax I cos
2
µ0 N 2 A Average power dissipated 2
Pav = Irms R
Inductance of solenoid L=
l
d B
Faraday’s law Eind =
dt
dI
Inductance Eind = L
Z dt
Magnetic potential energy UB = uB dV
Chapter 30
d E
Displacement current Idisp ⌘ ✏0
dt
1
EM wave speed c0 = p
✏ 0 µ0
I
Maxwell’s equations E ⌘ E ~ = qenc
~ · dA
I ✏0
B ⌘ B~ · dA
~=0
I
~ · d~` = d B
E
I dt
~ · d~` = µ0 I + µ0 ✏0 d
B
E
dt
Poynting vector ~⌘ 1E
S ~ ⇥B
~
µ0
Z
Power transported by EM wave P = ~ · dA
S ~
surface
Magnitudes E0 = c 0 B 0
Chapter 32
1
Capacitive reactance XC ⌘
!C ω= 2 π f
Inductive reactance XL ⌘ !L
Voltage across capacitor VC = IXC ow
freq fc
:
: 真
2π C
filter
Voltage across inductor VL = IXL
s ✓ ◆2
1
Impedance ZRLC ⌘ R2 + !L
!C
VR R
Power factor cos = =
Emax Z
Emax
Current amplitude I=
Z
3
19
Magnitude of electron charge e = 1.60 ⇥ 10 C q1 q2
Coulomb’s Law F~12
E
= k 2 r̂12
r12
Coulomb’s constant k = 8.99 ⇥ 109 N.m2 C 2
~r2 ~r1
Directional unit vector r̂12 =
1 r12
Permittivity of free space ✏0 =
4⇡k
12
✏0 = 8.85 ⇥ 10 C2 N 1
m 2
Permeability of free space µ0 = 4⇡ ⇥ 10 7
H.m 1 Chapter 23
31
Mass of electron me = 9.11 ⇥ 10 kg
Dipole moment (electric) p~ ⌘ qp~rp
Speed of light c0 = 3.0 ⇥ 10 m/s 8 P ~
Torque on electric dipole ~⌧ = p~ ⇥ E
~E
Electric field ~ ⌘ Ft
E
Mathematics qt
p
b± b2 4ac Induced dipole moment ~
p~ind = ↵E
2
Quadratic ax + bx + c = 0 x=
2a
Superposition of electric fields ~ =E
E ~1 + E
~2 + · · ·
~ =A
Adding vectors C ~+B
~ C x = Ax + B x
q
Uniform linear charge density ⌘
C y = Ay + B y `
q
~ = Ax î + Ay ĵ + Az k̂ Uniform surface charge density ⌘
Vector components A a
q
Uniform volume charge density ⇢⌘
✓ counterclockwise from x-axis Ax = A cos ✓, Ay = A sin ✓ V
~·B
~ = AB cos ✓ Electric field due to:
Scaler(Dot) product A
A point charge ~ s = k qs r̂sP
E
~·B
A ~ = Ax B x + Ay B y 2
8rsP
>
>
Vector(Cross) product ~ ⇥ B|
|A ~ = AB sin ✓ >
< 2kp if on y-axis
y3 ,
Dipole (aligned with y-axis, Ey ⇡
~⇥B
~ = ~ ⇥A
~ >
>
A B >
: kp
far from dipole) |x3 | , if on x-axis
Area sphere 4⇡r2
2k
An infinite line of charge Ex =
4 3 x
Volume sphere 3 ⇡r
s qz
A charged ring (on the axis) Ez = k
⌃ni=1 (xi xave )
2 (z 2 + R2 )3/2
Standard deviation =
n 1 1 1
A charged disk (on the axis) Ez = 2k⇡ z
|z| (z + R2 )1/2
2
Equations from 121 An infinite plane Ez = 2k⇡
8
>
>
>
<k rq2 , if r > R
Constant acceleration (x dir.) xf = xi + vx,i t + 12 ax t2
A thin spherical shell E=
>
>
vx,f = vx,i + ax t >
:0, if r < R
2 2
vx,f = vx,i + 2ax x
x = 12 (vx,i + vx,f ) t
Chapter 24
Kinetic energy K = 12 mv 2
P~ Z
F Electric flux = ~ · dA
E ~
Equation of motion ~a = E
m I
Interaction pair F12 = F~21
~ Gauss’s law E = E ~ = qenc
~ · dA
✏0
1
,Chapter 25 Chapter 14
q1 q2 1
Electric potential energy UE = `proper
4⇡✏0 r12 Z Length contraction `v =
B
Electrostatic work Wq (A ! B) = q ~ · d~`
E 1
A Lorentz factor ⌘q
v2
Potential (0 at 1) due to: 1 c20
1 X qn Time dilation tv = tproper
Point charges Vp =
4⇡✏0 rnP
1 q Lorenta trans frumation
A charged ring (on the axis) V =
tuew =
Z (t -
最 x ]
4⇡✏0 (z + R2 )1/2
2
⇣p ⌘ xew =
z ( x rt }
-
A charged disk (on the axis) V = z 2 + R2 |z|
2✏0 Chapter 28
1 q
A charged sphere (r > R) V =
4⇡✏0 r I
~ @V @V @V ~ · d~` = µ0 Ienc
Electric field (from potential) E= î ĵ k̂ Ampere’s law B
@x @y @z
~
Potential di↵erence VAB ⌘
Wq (A ! B)
Biot-Savart law ~ s = µ0 Id` ⇥ r̂sP
dB
4⇡ 2
rsP
q
I Z
Potential di↵erence closed path ~ · d~` = 0
E ~ =
B dB~s
current path
µ0 I
Magnetic field due to long wire B=
Chapter 26 2⇡r
Magnetic field due to solenoid B = µ0 nI
q
Capacitance C⌘ µ0 N I
Vcap Magnetic field due to toroid B=
✏0 A 2⇡r
Parallel plate capacitor C=
d
2⇡✏0 l
Coaxial cylindrical capacitor C=
ln(R2 /R1 )
V0
Dielectric constant ⌘ Chapter 31
Vd
Wnonelectrostatic
Emf E⌘
q J
2 Conductivity ⌘
Capacitor potential energy U E
= 1q
= 1 2 1 E
2 C 2 CVcap = 2 qVcap
ne2 ⌧
1 Conductivity metal =
Energy density uE = ✏0 E 2 me
2 l
Resistance R=
A
Chapter 27 |I|
Current Density J⌘
A
dq ~
eE
Current I⌘ Drift velocity ~vd = ⌧
dt me
⇣ ⌘
Electromagnetic force F~PEB = q E~ + ~v ⇥ B
~
Equiv. Resistance (series) Req = R1 + R2 + R3 + · · ·
I
Gauss’s law for magnetism = ~ · dA
B ~=0 1 1 1 1
B (parallel) = + + + ···
Req R1 R2 R3
B
~ Fw, max
Magnetic field (wire ? B) B⌘ Junction rule Iin = Iout
|I|`
Z
~ · dA
~ X X
Magnetic flux B = B Loop rule E+ V =0
Magnetic force F~wB = I ~` ⇥ B
~ = q~v ⇥ B
~ V
Ohm’s law I=
R
mv V
Particle in magnetic field R= Resistance R⌘
|q|B I
V2
Hall probe voltage across w V = vwB Power dissipated by a resistor P = V I = I 2R =
R
2
, Chapter 29 1
Resonant angular frequency !0 = p
LC
2π
=
f
1 B2 Xmax
Energy density uB ⌘ Rms Xrms = p
2 µ0 2
1
Inductor potential energy U B = 12 LI 2 Average power delivered Pav = Emax I cos
2
µ0 N 2 A Average power dissipated 2
Pav = Irms R
Inductance of solenoid L=
l
d B
Faraday’s law Eind =
dt
dI
Inductance Eind = L
Z dt
Magnetic potential energy UB = uB dV
Chapter 30
d E
Displacement current Idisp ⌘ ✏0
dt
1
EM wave speed c0 = p
✏ 0 µ0
I
Maxwell’s equations E ⌘ E ~ = qenc
~ · dA
I ✏0
B ⌘ B~ · dA
~=0
I
~ · d~` = d B
E
I dt
~ · d~` = µ0 I + µ0 ✏0 d
B
E
dt
Poynting vector ~⌘ 1E
S ~ ⇥B
~
µ0
Z
Power transported by EM wave P = ~ · dA
S ~
surface
Magnitudes E0 = c 0 B 0
Chapter 32
1
Capacitive reactance XC ⌘
!C ω= 2 π f
Inductive reactance XL ⌘ !L
Voltage across capacitor VC = IXC ow
freq fc
:
: 真
2π C
filter
Voltage across inductor VL = IXL
s ✓ ◆2
1
Impedance ZRLC ⌘ R2 + !L
!C
VR R
Power factor cos = =
Emax Z
Emax
Current amplitude I=
Z
3
