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PHYSICS FORMULAS
Electron = -1.602 19 × 10-19^ C = 9.11 × 10-31^ kg
Proton = 1.602 19 × 10-19^ C = 1.67 × 10-27^ kg
Neutron = 0 C = 1.67 × 10-27^ kg
6.022 × 10^23 atoms in one atomic mass unit
e is the elementary charge: 1.602 19 × 10-19^ C
Potential Energy, velocity of electron: PE = eV = ½ mv^2
1V = 1J/C 1N/C = 1V/m 1J = 1 N·m = 1 C·V
1 amp = 6.21 × 10^18 electrons/second = 1 Coulomb/second
1 hp = 0.756 kW 1 N = 1 T·A·m 1 Pa = 1 N/m^2
Power = Joules/second = I^2 R = IV [watts W ]
Quadratic
Equation: x
b b ac
a
Kinetic Energy [J]
KE = 12 mv^2
[Natural Log: when e b^ = x , ln x = b ] m: 10-3^ μ: 10-6^ n: 10-9^ p: 10-12^ f: 10-15^ a: 10-
Addition of Multiple Vectors:
r r r r R r = A r + B +r C r Resultant = Sum of the vectors R (^) x = A (^) x + B (^) x + Cx x -component A (^) x = A cos θ r r r r R (^) y = A (^) y + B (^) y + Cy y -component A (^) y = A sin θ
R = R x^2 + Ry^2 Magnitude (length) of R
θR
y x
R
R
= tan− 1 or tan θR
y x
R
R
= Angle of the resultant
Multiplication of Vectors:
Cross Product or Vector Product:
i × j = k j × i = − k i × i = 0
Positive direction:
i
j k
Dot Product or Scalar Product:
i ⋅ j = 0 i ⋅ i = 1 a ⋅ b = (^) ab cos θ k
i
j
Derivative of Vectors:
Velocity is the derivative of position with respect to time:
v = + + k = i + j + k
d
dt
x y z
dx
dt
dy
dt
dz
dt
( i j )
Acceleration is the derivative of velocity with respect to
time:
a = + + k = i + j + k
d
dt
v v v
dv
dt
dv
dt
dv x y z dt
x y^ z ( i j )
Rectangular Notation: (^) Z = R ± jX where + j represents
inductive reactance and - j represents capacitive reactance.
For example, Z = 8 + j 6 Ω means that a resistor of 8Ω is in series with an inductive reactance of 6Ω. Polar Notation: Z = M ∠ θ , where M is the magnitude of the
reactance and θ is the direction with respect to the
horizontal (pure resistance) axis. For example, a resistor of 4 Ω in series with a capacitor with a reactance of 3Ω would be expressed as 5 ∠-36.9° Ω. In the descriptions above, impedance is used as an example. Rectangular and Polar Notation can also be used to express amperage, voltage, and power.
To convert from rectangular to polar notation:
Given: X - j Y (careful with the sign before the ”j”)
Magnitude: X^2 + Y^2 = M
Angle:
tan θ =
− Y X
(negative sign carried over from rectangular notation in this example)
Note: Due to the way the calculator works, if X is negative,
you must add 180° after taking the inverse tangent. If the result is greater than 180°, you may optionally subtract 360° to obtain the value closest to the reference angle.
To convert from polar to rectangular (j) notation:
Given: M ∠ θ X Value: M cosθ Y (j) Value: M sinθ
In conversions, the j value will have the same sign as the θ value for angles having a magnitude < 180°. Use rectangular notation when adding and subtracting. Use polar notation for multiplication and division. Multiply in polar notation by multiplying the magnitudes and adding the angles. Divide in polar notation by dividing the magnitudes and subtracting the denominator angle from the numerator angle.
X
Magnitude M θ
Y
ELECTRIC CHARGES AND FIELDS
Coulomb's Law: [Newtons N ]
F k
q q
r
where: F = force on one charge by
the other[ N ]
k = 8.99 × 10^9 [ N · m^2 / C^2 ]
q 1 = charge [ C ]
q 2 = charge [ C ]
r = distance [ m ]
Electric Field: [Newtons/Coulomb or Volts/Meter]
E k
q
r
F
q
= 2 =
where: E = electric field [ N/C or V/m ]
k = 8.99 × 10^9 [ N · m^2 / C^2 ]
q = charge [ C ]
r = distance [ m ]
F = force
Electric field lines radiate outward from
positive charges. The electric field
is zero inside a conductor.
Relationship of k to ∈∈ 0 :
k = ∈
1
4 π 0
where: k = 8.99 × 10^9 [ N · m^2 / C^2 ]
∈ 0 = permittivity of free space 8.85 × 10-12^ [C^2 /N·m^2 ]
Electric Field due to an Infinite Line of Charge: [ N/C ]
E r
k
r
= ∈
=
λ
π
λ
2
2
0
E = electric field [ N/C ]
λ = charge per unit length [ C/m }
∈ 0 = permittivity of free space 8.85 × 10-12^ [C^2 /N·m^2 ]
r = distance [ m ]
k = 8.99 × 10^9 [ N · m^2 / C^2 ]
Electric Field due to ring of Charge: [ N/C ]
E
kqz
z R
= ( + )
2 2 3 2/
or if z >> R, E
kq
z
= (^2)
E = electric field [ N/C ]
k = 8.99 × 10^9 [ N · m^2 / C^2 ]
q = charge [ C ]
z = distance to the charge [ m ]
R = radius of the ring [ m ]
Electric Field due to a disk Charge: [ N/C ]
E
z
z R
= ∈
−
σ
2
1 0
2 2
E = electric field [ N/C ]
σ = charge per unit area
[ C/m^2 } ∈ 0 = 8.85 × 10-12^ [C^2 /N·m^2 ]
z = distance to charge [ m ]
R = radius of the ring [ m ]
Electric Field due to an infinite sheet: [ N/C ]
E = ∈
σ
(^2 )
E = electric field [ N/C ]
σ = charge per unit area [ C/m^2 }
∈ 0 = 8.85 × 10-12^ [C^2 /N·m^2 ]
Electric Field inside a spherical shell: [ N/C ]
E
kqr
R
= (^3)
E = electric field [ N/C ]
q = charge [C]
r = distance from center of sphere to
the charge [ m ]
R = radius of the sphere [ m ]
Electric Field outside a spherical shell: [ N/C ]
E
kq
r
= (^2)
E = electric field [ N/C ]
q = charge [C]
r = distance from center of sphere to
the charge [ m ]
Average Power per unit area of an electric or
magnetic field:
W m
E
c
m Bm^ c / 2
2
0
2
(^2 )
= = μ μ
W = watts Em = max. electric field [ N/C ] μ 0 = 4π × 10-
c = 2.99792 × 10^8 [ m/s ]
Bm = max. magnetic field [ T ]
A positive charge moving in the same direction as the electric field direction loses potential energy since the potential of the electric field diminishes in this direction. Equipotential lines cross EF lines at right angles.
Electric Dipole: Two charges of equal magnitude and opposite polarity separated by a distance d.
z
-Q
p
d +Q
E
k
z
=
2 3
p
E z
= ∈
1
2 π 0 3
p
when z » d
E = electric field [ N/C ]
k = 8.99 × 10^9 [ N · m^2 / C^2 ]
∈ 0 = permittivity of free space 8.85 × 10 -12^ C^2 / N · m^2
p = qd [ C · m ] "electric dipole moment"
in the direction negative to positive
z = distance [m] from the dipole
center to the point along the dipole axis where the electric field is to be measured
Deflection of a Particle in an Electric Field:
2 ymv^2 = qEL^2
y = deflection [ m ]
m = mass of the particle [ kg ]
d = plate separation [ m ]
v = speed [ m/s ]
q = charge [ C ]
E = electric field [ N/C or V/m
L = length of plates [ m ]
Charge per unit Area: [ C/m^2 ]
q
A
σ = charge per unit area [ C/m^2 ]
q = charge [ C ] A = area [ m^2 ]
Energy Density: (in a vacuum) [ J/m^3 ]
u = 12 ∈ 0 E
(^2) u = energy per unit volume [ J/m^3 ] ∈ 0 = permittivity of free space 8.85 × 10-12^ C^2 / N·m^2 E = energy [ J ]
Capacitors in Series:
C eff C 1 C 2
Capacitors in Parallel:
Ceff = C 1 + C 2 ...
Capacitors connected in series all have the same charge q.
For parallel capacitors the total q is equal to the sum of the
charge on each capacitor.
Time Constant: [seconds]
τ = RC^ τ^ = time it takes the capacitor to reach 63.2%
of its maximum charge [seconds] R = series resistance [ohms Ω] C = capacitance [farads F ]
Charge or Voltage after t Seconds: [coulombs C ]
charging:
q Q ( e )
t
−
/ τ
V V S ( e )
t
−
/ τ
discharging:
q = Qe −^ t / τ
V = V Se −^ t / τ
q = charge after t seconds [coulombs C ] Q = maximum charge [coulombs C ] Q = CV e = natural log t = time [seconds] τ = time constant RC [seconds] V = volts [ V ] VS = supply volts [ V ]
[Natural Log: when e b^ = x , ln x = b ]
Drift Speed:
I ( )
Q
t
= = nqv Ad
∆ Q = # of carriers × charge/carrier ∆ t = time in seconds n = # of carriers q = charge on each carrier v d = drift speed in meters/second A = cross-sectional area in meters^2
RESISTANCE
Emf: A voltage source which can provide continuous current
[volts]
ε = IR + Ir ε^ = emf open-circuit voltage of the battery
I = current [amps] R = load resistance [ohms] r = internal battery resistance [ohms]
Resistivity: [Ohm Meters]
ρ =
E J
ρ =
RA L
ρ = resistivity [ Ω · m ]
E = electric field [ N/C ]
J = current density [ A/m^2 ]
R = resistance [ Ω ohms]
A = area [ m^2 ]
L = length of conductor [ m ]
Variation of Resistance with Temperature:
ρ − ρ (^) 0 = ρ (^) 0 α ( T − T 0 ) ρ = resistivity [ Ω · m ]
ρ 0 = reference resistivity [ Ω · m ]
α = temperature coefficient of
resistivity [ K-1 ]
T 0 = reference temperature
T - T 0 = temperature difference
[ K or ° C ]
CURRENT
Current Density: [ A/m^2 ]
i = ∫ J ⋅ d A
if current is uniform and parallel to d A, then: i = JA
J = ( ne V ) d
i = current [ A ]
J = current density [ A/m^2 ]
A = area [ m^2 ]
L = length of conductor [ m ]
e = charge per carrier
ne = carrier charge density [ C/m^3 ]
Vd = drift speed [ m/s ]
Rate of Change of Chemical Energy in a Battery:
P = iε P^ = power [ W ]
i = current [ A ]
ε = emf potential [ V ]
Kirchhoff’s Rules
- The sum of the currents entering a junctions is equal to the sum of the currents leaving the junction.
- The sum of the potential differences across all the elements around a closed loop must be zero.
Evaluating Circuits Using Kirchhoff’s Rules
- Assign current variables and direction of flow to all branches of the circuit. If your choice of direction is incorrect, the result will be a negative number. Derive equation(s) for these currents based on the rule that currents entering a junction equal currents exiting the junction.
- Apply Kirchhoff’s loop rule in creating equations for different current paths in the circuit. For a current path beginning and ending at the same point, the sum of voltage drops/gains is zero. When evaluating a loop in the direction of current flow, resistances will cause drops (negatives); voltage sources will cause rises (positives) provided they are crossed negative to positive—otherwise they will be drops as well.
- The number of equations should equal the number of variables. Solve the equations simultaneously.
MAGNETISM
André-Marie Ampére is credited with the discovery of
electromagnetism, the relationship between electric
currents and magnetic fields.
Heinrich Hertz was the first to generate and detect
electromagnetic waves in the laboratory.
Magnetic Force acting on a charge q : [Newtons N ]
F = qvB sin θ
F = q v × B
F = force [ N ] q = charge [ C ] v = velocity [ m/s ] B = magnetic field [ T ] θ = angle between v and B
Right-Hand Rule: Fingers represent the direction of the magnetic force B , thumb represents the direction of v (at any angle to B ), and the force F on a positive charge emanates from the palm. The direction of a magnetic field is from north to south. Use the left hand for a negative charge. Also, if a wire is grasped in the right hand with the thumb in the direction of current flow, the fingers will curl in the direction of the magnetic field. In a solenoid with current flowing in the direction of curled fingers, the magnetic field is in the direction of the thumb. When applied to electrical flow caused by a changing magnetic field , things get more complicated. Consider the north pole of a magnet moving toward a loop of wire (magnetic field increasing). The thumb represents the north pole of the magnet, the fingers suggest current flow in the loop. However, electrical activity will serve to balance the change in the magnetic field, so that current will actually flow in the opposite direction. If the magnet was being withdrawn, then the suggested current flow would be decreasing so that the actual current flow would be in the direction of the fingers in this case to oppose the decrease. Now consider a cylindrical area of magnetic field going into a page. With the thumb pointing into the page, this would suggest an electric field orbiting in a clockwise direction. If the magnetic field was increasing, the actual electric field would be CCW in opposition to the increase. An electron in the field would travel opposite the field direction (CW) and would experience a negative change in potential.
Force on a Wire in a Magnetic Field: [Newtons N ]
F = BIl sin θ
F = I l × B
F = force [ N ] B = magnetic field [ T ] I = amperage [ A ] l = length [ m ] θ = angle between B and the direction of the current
Torque on a Rectangular Loop: [Newton·meters N·m ]
τ = NBIA sin θ
N = number of turns B = magnetic field [ T ] I = amperage [ A ] A = area [ m^2 ] θ = angle between B and the plane of the loop
Charged Particle in a Magnetic Field:
r
mv
qB
=
r = radius of rotational path m = mass [ kg ] v = velocity [ m/s ] q = charge [ C ] B = magnetic field [ T ]
Magnetic Field Around a Wire: [T]
B
I
r
=
μ
π
0 2
B = magnetic field [ T ]
μ 0 = the permeability of free
space 4π×10-7^ T·m/A I = current [ A ] r = distance from the center of the conductor
Magnetic Field at the center of an Arc: [T]
B
i
r
=
μ φ
π
0 4
B = magnetic field [ T ]
μ 0 = the permeability of free
space 4π×10-7^ T·m/A i = current [ A ]
φ = the arc in radians
r = distance from the center of the conductor
Hall Effect: Voltage across the width of a conducting ribbon due to a Magnetic Field:
( ne V h ) w = Bi
v Bwd = Vw
ne = carrier charge density [ C/m^3 ]
Vw = voltage across the width [ V ] h = thickness of the conductor [ m ] B = magnetic field [ T ] i = current [ A ] vd = drift velocity [ m/s ] w = width [ m ]
Force Between Two Conductors: The force is
attractive if the currents are in the same direction.
F I I
d
1 0 1 2 l 2
=
μ
π
F = force [ N ] l = length [ m ]
μ 0 = the permeability of free
space 4π×10-7^ T·m/A I = current [ A ] d = distance center to center [ m ]
Magnetic Field Inside of a Solenoid: [Teslas T ] B = μ 0 nI^ B^ = magnetic field [ T ]
μ 0 = the permeability of free
space 4π×10-7^ T·m/A n = number of turns of wire per unit length [ #/m ] I = current [ A ]
Magnetic Dipole Moment: [ J / T ]
μ = NiA μ^ = the magnetic dipole moment [ J / T ] N = number of turns of wire i = current [ A ] A = area [ m^2 ]
Magnetic Flux through a closed loop: [ T · M^2 or Webers]
Φ = BA cos θ B^ = magnetic field [ T ] A = area of loop [ m^2 ] θ = angle between B and the perpen-dicular to the plane of the loop
Voltage, series circuits: [V]
V
q
C
C =^ V^ R = IR
V
X
V
R
I
X R = =
V^2 = VR^2 + VX^2
V C = voltage across capacitor [ V ] q = charge on capacitor [ C ] fR = Resonant Frequency [ Hz ] L = inductance [ H ] C = capacitance in farads [ F ] R = resistance [ Ω ] I = current [ A ] V = supply voltage [ V ] V X = voltage across reactance [ V ] V R = voltage across resistor [ V ]
Phase Angle of a series RL or RC circuit: [degrees]
tan φ = =
X
R
V
V
X
R
cos φ = =
V
V
R
Z
R
( φ would be negative in a capacitive circuit)
φ = Phase Angle [degrees] X = reactance [ Ω ] R = resistance [ Ω ] V = supply voltage [ V ] V X = voltage across reactance [ V ] V R = voltage across resistor [ V ] Z = impedance [ Ω ]
Impedance of a series RL or RC circuit: [ Ω ]
Z^2 = R^2 + X^2 E = I Z Z
V
X
V
R
V
C
C R
= =
Z = R ± j X
φ = Phase Angle [degrees] X = reactance [ Ω ] R = resistance [ Ω ] V = supply voltage [ V ] V X = voltage across reactance [ V ] V X = voltage across resistor [ V ] Z = impedance [ Ω ]
Series RCL Circuits:
The Resultant Phasor X = X (^) L − XC is
in the direction of the larger reactance and determines whether the circuit is inductive or capacitive. If XL is larger than XC , then the circuit is inductive and X is a vector in the upward direction.
In series circuits, the amperage is the reference (horizontal) vector. This is observed on the oscilloscope by looking at the voltage across the resistor. The two vector diagrams at right illustrate the phase relationship between voltage, resistance, reactance, and amperage.
XC
XL
I
R
VL
VC
I
V R
Series RCL
Impedance Z^ R^ X^ L^ XC
(^2) = 2 + ( − ) (^2) Z
R
cos φ Impedance may be found by adding the components using vector algebra. By converting the result to polar notation, the phase angle is also found. For multielement circuits, total each resistance and reactance before using the above formula.
Damped Oscillations in an RCL Series Circuit:
q = Qe −^ Rt^ /^2 L cos( ω ′ + t φ )
where
ω ′ = ω −
2 2 ( R / 2 L )
ω = 1 / LC
When R is small and ω′ ≈ ω:
U
Q C
= e − Rt L
2
2
/
q = charge on capacitor [ C ] Q = maximum charge [ C ] e = natural log R = resistance [ Ω ] L = inductance [ H ] ω = angular frequency of the undamped oscillations [ rad/s ] ω = angular frequency of the damped oscillations [ rad/s ] U = Potential Energy of the capacitor [ J ] C = capacitance in farads [ F ]
Parallel RCL Circuits:
I (^) T = I (^) R + I (^) C − IL
2 2 ( )
tan φ =
I − I
I
C L
R
V
IL
I
R
C
I
To find total current and phase angle in multielement circuits, find I for each path and add vectorally. Note that when converting between current and resistance, a division will take place requiring the use of polar notation and resulting in a change of sign for the angle since it will be divided into (subtracted from) an angle of zero.
Equivalent Series Circuit: Given the Z in polar notation of a
parallel circuit, the resistance and reactance of the equivalent series circuit is as follows: R = ZT cos θ X = ZT sin θ
AC CIRCUITS
Instantaneous Voltage of a Sine Wave:
V = V max sin 2π ft
V = voltage [ V ] f = frequency [ Hz ] t = time [s]
Maximum and rms Values:
I
Im
2
V
Vm
2
I = current [ A ] V = voltage [ V ]
RLC Circuits: V = V (^) R^2 + ( V (^) L − VC )^2 Z = R^2 + ( X (^) L − XC )^2
tan φ =
X − X
R
L C Pavg^ =^ IV cos φ PF = cos φ
Conductance (G): The
reciprocal of resistance in siemens (S).
Susceptance (B, BL, BC): The
reciprocal of reactance in siemens (S).
Admittance (Y): The reciprocal
of impedance in siemens (S).
B Y
Susceptance Conductance
Admittance G
ELECTROMAGNETICS
WAVELENGTH
c =λ f
c = E / B
1Å = 10
c = speed of light 2.998 × 10^8 m/s λ = wavelength [m]
f = frequency [Hz]
E = electric field [N/C]
B = magnetic field [T]
Å = (angstrom) unit of wavelength
equal to 10-10^ m
m = (meters)
WAVELENGTH SPECTRUM
BAND METERS ANGSTROMS Longwave radio (^) 1 - 100 km 1013 - 10^15
Standard Broadcast (^) 100 - 1000 m 1012 - 10^13 Shortwave radio (^) 10 - 100 m 1011 - 10^12 TV, FM (^) 0.1 - 10 m 109 - 10^11
Microwave (^) 1 - 100 mm 107 - 10^9 Infrared light (^) 0.8 - 1000 μm 8000 - 10^7
Visible light (^) 360 - 690 nm 3600 - 6900 violet (^) 360 nm 3600
blue (^) 430 nm 4300 green (^) 490 nm 4900 yellow (^) 560 nm 5600
orange (^) 600 nm 6000 red (^) 690 nm 6900 Ultraviolet light (^) 10 - 390 nm 100 - 3900
X-rays (^) 5 - 10,000 pm 0.05 - 100 Gamma rays (^) 100 - 5000 fm 0.001 - 0. Cosmic rays (^) < 100 fm < 0.
Intensity of Electromagnetic Radiation [watts/m^2 ]:
I
P
r
s
4 π 2
I = intensity [ w/m^2 ] Ps = power of source [watts]
r = distance [ m ]
4 πr^2 = surface area of sphere
Force and Radiation Pressure on an object:
a) if the light is totally
absorbed:
F
IA
c
= P
I r (^) c =
b) if the light is totally reflected back along the path:
F
IA
c
=
2 P
I r (^) c =
2
F = force [ N ] I = intensity [ w/m^2 ] A = area [ m^2 ] Pr = radiation pressure [ N/m^2 ]
c = 2.99792 × 10^8 [ m/s ]
Poynting Vector [watts/m^2 ]:
S = EB = E
1 1
0 0
2 μ μ
cB = E
μ 0 = the permeability of free
space 4π×10-7^ T·m/A E = electric field [ N/C or V/M ] B = magnetic field [ T ]
c = 2.99792 × 10^8 [ m/s ]
LIGHT
Indices of Refraction: Quartz:^ 1. Glass, crown 1. Glass, flint 1. Water 1. Air 1.000 293
Angle of Incidence: The angle measured from the perpendicular to the face or from the perpendicular to the tangent to the face
Index of Refraction: Materials of greater density have
a higher index of refraction.
n
c
v
≡
n = index of refraction c = speed of light in a vacuum 3 × 10^8 m/s v = speed of light in the material [ m/s ]
n n
=
λ λ
(^0) λ 0 = wavelength of the light in a vacuum [ m ] λν = its wavelength in the material [ m ]
Law of Refraction: Snell’s Law n (^) 1 sin θ 1 (^) = n 2 sin θ 2 n = index of refraction θ = angle of incidence traveling to a region of lesser density: θ (^) 2 > θ 1
refracted
Source
n
θ
nn (^1 )
2
1
θ (^2)
traveling to a region of greater density: θ 2 (^) < θ 1
refracted
Source
n
θ
nn (^1 )
2
1
θ (^2)
Critical Angle: The maximum angle of incidence for which light
can move from n 1 to n 2
sin θ c
n n
= 2 1
for n 1 > n 2
Sign Conventions: When M is negative, the image is inverted. p is positive when the object is in front of the mirror, surface, or lens. Q is positive when the image is in front of the mirror or in back of the surface or lens. f and r are positive if the center of curvature is in front of the mirror or in back of the surface or lens. Magnification by spherical mirror or thin lens. A
negative m means that the image is inverted.
M
h h
i p
=
′ = −
h’ = image height [ m ] h = object height [ m ] i = image distance [ m ] p = object distance [ m ]
reflected
refracted
θ
Source
θ
nn (^11)
n (^2)
