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**PHYSICS FORMULAS
**2426

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 × 1023 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 × 1018 electrons/second = 1 Coulomb/second 1 hp = 0.756 kW 1 N = 1 T·A·m 1 Pa = 1 N/m2

Power = Joules/second = *I*2*R* = *IV* [watts W]
Quadratic
Equation: *x
*

*b b ac
*

*a
*=

− ± −2 4 2

Kinetic Energy [J]

*KE mv*= 12
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: 10-18

Addition of Multiple Vectors:

r r r r
*R A B C*= + + Resultant = Sum of the vectorsr r r r
*R A B Cx x x x*= + + x-component *A Ax *= cos θr r r r
*R A B Cy y y y*= + + y-component *A Ay *= sin θ

*R R Rx y*= +
2 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
*

*dtx y z
x y z*( )**i j
**

Rectangular Notation: *Z R jX*= ± where +*j* represents
inductive reactance and -*j* represents capacitive reactance.
For example, *Z j*= +8 6Ω 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
4Ω 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 Y M*2 2+ =
Angle:

tanθ =
− *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
*

*M
*Ma

gni tud

e

*θ
Y*

**ELECTRIC CHARGES AND FIELDS
**

Coulomb's Law: [Newtons N]

*F k
q q
*

*r
*= 1 2

2

where: *F* = force on one charge by
the other[N]

*k* = 8.99 × 109 [N·m2/C2]
*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 × 109 [N·m2/C2]
*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 0*π
*

where: *k* = 8.99 × 109 [N·m2/C2]
∈0 = permittivity of free space

8.85 × 10-12 [C2/N·m2]

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 [C2/N·m2]
*r* = distance [m]
*k* = 8.99 × 109 [N·m2/C2]

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 × 109 [N·m2/C2]
*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/m2}
∈0 = 8.85 × 10-12 [C2/N·m2]
*z* = distance to charge [m]
*R* = radius of the ring [m]

Electric Field due to an infinite sheet: [N/C]

*E *=
∈
*σ
*

2 0

*E* = electric field [N/C]
*σ* = charge per unit area [C/m2}
∈0 = 8.85 × 10-12 [C2/N·m2]

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
*

*B cm m*/ 2
2

0

2

02 2 = =

µ µ

*W* = watts
*Em* = max. electric field [N/C]
*µ0* = 4π × 10-7

*c* = 2.99792 × 108 [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 0
3*π
*

**p
**

when *z* » *d
*

*E* = electric field [N/C]
*k* = 8.99 × 109 [N·m2/C2]
∈0 = permittivity of free space 8.85 ×

10-12 C2/N·m2

**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 2 2*ymv qEL*= *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]

Potential Difference between two Points: [volts V]

∆ ∆

*V V V
PE
*

*q
EdB A*= − = = −

∆*PE* = work to move a charge
from A to B [N·m or J]

*q* = charge [C]
*V*B = potential at B [V]
*V*A = potential at A [V]
*E* = electric field [N/C or V/m
*d* = plate separation [m]

Electric Potential due to a Point Charge: [volts V]

*V k
q
*

*r
*=

*V* = potential [volts V]
*k* = 8.99 × 109 [N·m2/C2]
*q* = charge [C]
*r* = distance [m]

Potential Energy of a Pair of Charges: [J, N·m or C·V]

*PE q V k
q q
*

*r
*= =2 1

1 2
*V*1 is the electric potential due to

q1 at a point P
*q*2*V*1 is the work required to bring

q2 from infinity to point P

Work and Potential:

∆*U U U Wf i*= − = −

*U W*= − ∞
*W Fd*= ⋅ =**F d **cos*θ
*

*W q d
i
*

*f
*

= ⋅∫ **E s
**

∆*V V V
W
*

*qf i
*= − = −

*V d
i
*

*f
*

= − ⋅∫ **E s
**

*U* = electric potential energy [J]
*W* = work done on a particle by

a field [J]
*W∞* = work done on a particle

brought from infinity (zero potential) to its present location [J]

**F** = is the force vector [N]
**d** = is the distance vector over

which the force is applied[m]

*F* = is the force scalar [N]
*d* = is the distance scalar [m]
*θ* = is the angle between the

force and distance vectors
*d***s** = differential displacement of

the charge [m]
V = volts [V]
*q* = charge [C]

Flux: the rate of flow (of an electric field) [N·m2/C]

Φ = ⋅∫ **E A***d
*= ∫ *E dA*(cos )*θ
*

Φ is the rate of flow of an electric field [N·m2/C]

∫ integral over a closed surface
**E** is the electric field vector [N/C]
**A** is the area vector [m2] pointing

outward normal to the surface.

Gauss' Law:

∈ =0 Φ *qenc
*

∈ ⋅ =∫0 **E A***d qenc
*

∈0 = 8.85 × 10-12 [C2/N·m2] Φ is the rate of flow of an electric

field [N·m2/C]
*qenc* = charge within the gaussian

surface [C]

∫ integral over a closed surface
**E** is the electric field vector [J]
**A** is the area vector [m2] pointing

outward normal to the surface.

**CAPACITANCE
**

Parallel-Plate Capacitor:

*C
A
*

*d
*= ∈*κ *0

*C* = capacitance [farads F]

κ = the dielectric constant (1) ∈0 = permittivity of free space 8.85 × 10-12 C2/N·m2

*A* = area of one plate [m2]
*d* = separation between plates [m]

Cylindrical Capacitor:

*C
L
*

*b a
*= ∈2 0*πκ *ln( / )

*C* = capacitance [farads F]

κ = dielectric constant (1) ∈0 = 8.85 × 10

-12 C2/N·m2

*L* = length [m]
*b* = radius of the outer

conductor [m]
*a* = radius of the inner

conductor [m]

Spherical Capacitor:

*C
ab
*

*b a
*= ∈

−
4 0*πκ
*

*C* = capacitance [farads F]

κ = dielectric constant (1) ∈0 = 8.85 × 10

-12 C2/N·m2

*b* = radius, outer conductor
[m]

*a* = radius, inner conductor [m]

Maximum Charge on a Capacitor: [Coulombs C]

*Q VC*= Q = Coulombs [C]
V = volts [V]
C = capacitance in farads [F]

For capacitors connected in series, the charge Q is equal for each capacitor as well as for the total equivalent. If the dielectric constant κ is changed, the capacitance is multiplied by κ, the voltage is divided by κ, and Q is unchanged. In a vacuum κ = 1, When dielectrics are used, replace ∈0 with κ ∈0.

Electrical Energy Stored in a Capacitor: [Joules J]

*U
QV CV Q
*

*CE
*= = =

2 2 2

2 2 U = Potential Energy [J] Q = Coulombs [C] V = volts [V] C = capacitance in farads [F]

Charge per unit Area: [C/m2]

*σ *=
*q
*

*A
σ* = charge per unit area [C/m2]
q = charge [C]
A = area [m2]

Energy Density: (in a vacuum) [J/m3]

*u E*= ∈12 0
2 u = energy per unit volume [J/m3]

∈0 = permittivity of free space 8.85 × 10-12 C2/N·m2

E = energy [J]

**Capacitors in Series:
**

1 1 1

1 2*C C Ceff
*= + ...

**Capacitors in Parallel:
**

*C C Ceff *= +1 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*= − −1 /*τ
*( )*V V eS t*= − −1 /*τ
*

discharging:

*q Qe t*= − /*τ
*

*V V eS
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
vd = drift speed in meters/second
*A* = cross-sectional area in meters2

**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/m2]
*R* = resistance [*Ω* ohms]
*A* = area [m2]
*L* = length of conductor [m]

Variation of Resistance with Temperature:

*ρ ρ ρ α*− = −0 0 0( )*T T ρ* = resistivity [*Ω* · m]
*ρ0* = 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/m2]

*i d*= ⋅∫ **J A
**if current is uniform
and parallel to *d***A,
**

then: *i JA*=

*J ne Vd*= ( )

*i* = current [A]
*J* = current density [A/m2]
*A* = area [m2]
*L* = length of conductor [m]
*e* = charge per carrier
*ne* = carrier charge density [C/m3]
*Vd* = drift speed [m/s]

Rate of Change of Chemical Energy in a Battery:

*P i*= *ε P* = power [W]
*i* = current [A]
*ε* = emf potential [V]

Kirchhoff’s Rules 1. The sum of the currents entering a junctions is equal to

the sum of the currents leaving the junction. 2. The sum of the potential differences across all the

elements around a closed loop must be zero.

**Evaluating Circuits Using Kirchhoff’s Rules
**

1. 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.

2. 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.

3. 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*= sinθ

*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 BI*= *l*sin*θ
*

*F I B*= ×*l
*

*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 θ= *NBIA *sin *N* = number of turns
*B* = magnetic field [T]
*I* = amperage [A]
*A* = area [m2]
*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 Biw *=

*v Bw Vd w*=

*ne* = carrier charge density [C/m3]
*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

2*l
*=

*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 nI*= µ0
*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 [m2]

Magnetic Flux through a closed loop: [T·M2 or Webers]

Φ = *BA*cos*θ B* = magnetic field [T]
*A* = area of loop [m2]
*θ* = angle between *B* and the

perpen-dicular to the plane of the loop

Magnetic Flux for a changing magnetic field: [T·M2 or Webers]

Φ = ⋅∫ **B A***d
B* = magnetic field [T]
*A* = area of loop [m2]

A Cylindrical Changing Magnetic Field

**E s**⋅ = =∫ *d E r
d
*

*dt
B*2*π
*

Φ

Φ*B BA B r*= = *π
*2

*d
*

*dt
A
*

*dB
*

*dt
*

Φ =

*ε *= −*N
d
*

*dt
*

Φ

*E* = electric field [N/C]
*r* = radius [m]
*t* = time [s]
Φ = magnetic flux [T·m2 or

Webers]
*B* = magnetic field [T]
*A* = area of magnetic field

[m2]
*dB/dt* = rate of change of

the magnetic field [T/s]
*ε* = potential [V]
*N* = number of orbits

Faraday’s Law of Induction states that the instan-
taneous emf induced in a circuit equals the rate of
change of magnetic flux through the circuit. Michael
**Faraday** made fundamental discoveries in
magnetism, electricity, and light.

*ε *= − *N
t
*

∆Φ ∆

*N* = number of turns
Φ = magnetic flux [T·m2]
*t* = time [s]

Lenz’s Law states that the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through a circuit

Motional emf is induced when a conducting bar moves through a perpendicular magnetic field.

*ε *= *B v*l *B* = magnetic field [T]
*l* = length of the bar [m]
*v* = speed of the bar [m/s]

emf Induced in a Rotating Coil:

*ε ω ω*= *NAB t*sin *N* = number of turns
*A* = area of loop [m2]
*B* = magnetic field [T]
*ω* = angular velocity [rad/s]
*t* = time [s]

Self-Induced emf in a Coil due to changing current:

*ε *= − *L
I
*

*t
*

∆ ∆

*L* = inductance [H]
*I* = current [A]
*t* = time [s]

Inductance per unit length near the center of a solenoid:

*L
n A
*

l
= *µ*0

2
*L* = inductance [H]
*l* = length of the solenoid [m]
*µ*0 = the permeability of free space

4π×10-7 T·m/A
*n* = number of turns of wire per unit

length [#/m]
*A* = area [m2]

Amperes' Law:

**B s**⋅ =∫ *d iencµ*0
*B* = magnetic field [T]
*µ*0 = the permeability of free space

4π×10-7 T·m/A
*ienc* = current encircled by the

loop[A]

Joseph **Henry**, American physicist, made improvements
to the electromagnet.

James Clerk **Maxwell** provided a theory showing the
close relationship between electric and magnetic
phenomena and predicted that electric and magnetic
fields could move through space as waves.

J. J. **Thompson** is credited with the discovery of the
electron in 1897.

**INDUCTIVE & RCL CIRCUITS
**

Inductance of a Coil: [H]

*L
N
*

*I
*=

Φ *N* = number of turnsΦ = magnetic flux [T·m2]
*I* = current [A]

In an RL Circuit, after one time constant (*τ* = *L/R*) the
current in the circuit is 63.2% of its final value, ε/*R*.

RL Circuit:

current rise:

( )*I V
R
*

*e t L*= − −1 /*τ
*

current decay:

*I
V
*

*R
e t L*= − /*τ
*

UB = Potential Energy [J]
V = volts [V]
R = resistance [*Ω*]
*e* = natural log
*t* = time [seconds]
*τL* = inductive time constant *L/R
*

[s]
*I* = current [A]

Magnetic Energy Stored in an Inductor:

*U LIB *=
1
2

2 UB = Potential Energy [J]
*L* = inductance [H]
*I* = current [A]

Electrical Energy Stored in a Capacitor: [Joules J]

*U
QV CV Q
*

*CE
*= = =

2 2 2

2 2 UE = Potential Energy [J] Q = Coulombs [C] V = volts [V] C = capacitance in farads [F]

Resonant Frequency: : The frequency at which XL = XC.
In a **series**-resonant circuit, the impedance is at its
minimum and the current is at its maximum. For a
**parallel**-resonant circuit, the opposite is true.

*f
LC
*

*R *=
1

2*π
*

*ω *=
1

*LC
*

fR = Resonant Frequency [Hz]
*L* = inductance [H]
C = capacitance in farads [F]
*ω* = angular frequency [rad/s]

Voltage, series circuits: [V]

*V
q
*

*CC
*= *V IRR *=

*V
*

*X
*

*V
*

*R
IX R*= =

*V V VR X
*2 2 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 R X*2 2 2= +
*E I Z*=
*Z
*

*V
*

*X
*

*V
*

*R
*

*V
C
*

*C R
*

= =

*Z R X*= ± j

*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 XL C*= − 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
*

*CV
*

*I
*

*RV
*

Series RCL
Impedance *Z R X XL C
*

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 tRt L*= ′ +− / cos( )2 *ω φ
*

where

′ = −*ω ω *2 22( / )*R 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 I I IT R C L*= + −
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 ft*= max sin 2π
*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 V VR L C*= + −
2 2( ) *Z R X XL C*= + −

2 2( )

tan*φ *=
−*X X
R
*

*L C P IVavg *= 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).

*YB
*

Su sc

ep ta

nc e

Conductance

Ad mi

ttan ce

*G*

**ELECTROMAGNETICS
**

**WAVELENGTH
**

*fc *λ=
*BEc */=

1Å = 10-10m

*c* = speed of light 2.998 × 108 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 - 1015

Standard Broadcast 100 - 1000 m 1012 - 1013

Shortwave radio 10 - 100 m 1011 - 1012

TV, FM 0.1 - 10 m 109 - 1011

Microwave 1 - 100 mm 107 - 109

Infrared light 0.8 - 1000 µm 8000 - 107

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 - 0.05 Cosmic rays < 100 fm < 0.001

Intensity of Electromagnetic Radiation [watts/m2]:

*I
P
*

*r
s*=

4 2*π
*

*I* = intensity [w/m2]
*Ps* = power of source [watts]
*r* = distance [m]
4*π*r2 = surface area of sphere

Force and Radiation Pressure on an object:

a) if the light is totally absorbed:

*F
IA
*

*c
*= *P
*

*I
*

*cr
*=

b) if the light is totally reflected back along the path:

*F
IA
*

*c
*=

2
*P
*

*I
*

*cr
*=

2

*F* = force [N]
*I* = intensity [w/m2]
*A* = area [m2]
*Pr* = radiation pressure [N/m

2]
*c* = 2.99792 × 108 [m/s]

Poynting Vector [watts/m2]:

*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 × 108 [m/s]

**LIGHT
**

Indices of Refraction: Quartz: 1.458 Glass, crown 1.52 Glass, flint 1.66 Water 1.333 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 × 108 m/s
*v* = speed of light in the material [m/s]

*n
n
*

= λ λ

0 *λ0* = wavelength of the light in a vacuum [m]
*λν* = its wavelength in the material [m]

Law of Refraction: **Snell’s Law
***n n*1 1 2 2sin sin*θ θ*= *n* = index of refraction

*θ* = angle of incidence
traveling to a region of
lesser density: *θ θ*2 1>

*refracted
*

*Source
*

*n
*

*θ
n1
*

*2
*

*1
*

*θ2
*

traveling to a region of
greater density:
*θ θ*2 1<

*refracted
*

*Source
*

*n
*

*θ
n1
*

*2
*

*1
*

*θ2
*

Critical Angle: The maximum
angle of incidence for which light
can move from *n1* to *n2
*

sinθ*c
n
*

*n
*= 2

1

for *n1* > *n2
*

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
*

*n1
*

*n2*

Plane Refracting Surface: plane refracting surface:

*n
*

*p
*

*n
*

*i
*1 2= −

*p* = object distance
*i* = image distance [m]
*n* = index of refraction

Lensmaker’s Equation for a thin lens in air:

( )1 1 1 1 1 1
1 2*f p i
*

*n
r r
*

= + = − −

*r1* = radius of surface nearest the
object[m]

*r2* = radius of surface nearest the
image [m]

*f* = focal length [m]
*i* = image distance [m]
*p* = object distance [m]
*n* = index of refraction

Virtual Image

*C2
*

*r2
*

*F1
*

*r1
*

*F2 C1
*

*C2
*

*F1
*

*p
*

*F2
*

*i
*

*C1
*

Real Image

Thin Lens when the thickest part is thin compared to p.
*i* is negative on the left, positive on the right

*f
r
*

= 2

*f* = focal length [m]
*r* = radius [m]

Converging Lens

*f* is positive (left)
*r1* and *r2* are positive in

this example

Diverging Lens

*f* is negative (right)
*r1* and *r2* are negative in

this example

Two-Lens System Perform the calculation in steps. Calculate the image produced by the first lens, ignoring the presence of the second. Then use the image position relative to the second lens as the object for the second calculation ignoring the first lens.

Spherical Refracting Surface This refers to two materials with a single refracting surface.

*n
*

*p
*

*n
*

*i
*

*n n
*

*r
*1 2 2 1+ =

−

*M
h
*

*h
*

*n i
*

*n p
*=

′ = − 1

2

*p* = object distance
*i* = image distance [m] (positive for real

images)
*f* = focal point [m]
*n* = index of refraction
*r* = radius [m] (positive when facing a

convex surface, unlike with mirrors)
*M* = magnification
*h'* = image height [m]
*h* = object height [m]

Constructive and Destructive Interference by Single and Double Slit Defraction and Circular Aperture Young’s double-slit experiment (bright fringes/dark fringes):

Double Slit
Constructive:
∆*L d m*= =sin*θ λ
*

Destructive:

∆*L d m*= = +sin ( )*θ λ*12

*d* = distance between the slits [m]
*θ* = the angle between a normal

line extending from midway between the slits and a line extending from the midway point to the point of ray

Intensity:

*I Im*=

(cos )

sin2 2

*β
α
*

*α
*

*β
π
λ
*

*θ*=
*d
*

sin

*α
π
λ
*

*θ*=
*a
*

sin

Single-Slit
Destructive:
*a m*sin*θ λ*=

Circular Aperture 1st Minimum:

sin . .

*θ
γ
*

= 122
*dia
*

intersection.
*m* = fringe order number [integer]
*λ* = wavelength of the light [m]
*a* = width of the single-slit [m]
*∆L* = the difference between the

distance traveled of the two rays [m]

*I* = intensity @ *θ* [W/m2]
*Im* = intensity @ *θ* = 0 [W/m2]
*d* = distance between the slits [m]

In a circular aperture, the 1st

minimum is the point at which an image can no longer be resolved.

A **reflected ray** undergoes a **phase shift** of 180°
when the reflecting material has a greater index of
refraction n than the ambient medium. Relative to the
same ray without phase shift, this constitutes a path
difference of λ/2.

Interference between Reflected and Refracted rays
from a thin material **surrounded** by another medium:
Constructive:

2 12*nt m*= +( )λ
Destructive:

2*nt m*= λ

*n* = index of refraction
*t* = thickness of the material [m]
*m* = fringe order number [integer]
*λ* = wavelength of the light [m]

If the thin material is between two different media, one with a higher n and the other lower, then the above constructive and destructive formulas are reversed.

Wavelength within a medium:

λ λ

*n n
*=

*c n fn*= *λ
*

*λ* = wavelength in free space [m]
*λn* = wavelength in the medium [m]
*n* = index of refraction
*c* = the speed of light 3.00 × 108 [m/s]
*f* = frequency [Hz]

Polarizing Angle: by **Brewster’s Law**, the angle of
incidence that produces complete polarization in the
reflected light from an amorphous material such as glass.

tan*θB
n
*

*n
*= 2

1

*θ θr B*+ = °90

*n* = index of refraction
*θB* = angle of incidence

producing a 90° angle between reflected and refracted rays.

*θr* = angle of incidence of the
refracted ray.

*partially polarized
*

*θr
*

*n
*

*2
*

*1
*

*n
*

*θbθb
*

*non-polarized
Source
*

*polarized
*

Intensity of light passing through a polarizing lense: [Watts/m2]

initially unpolarized: *I I*= 12 0
initially polarized:

*I I*= 0
2cos *θ
*

*I* = intensity [W/m2]
*I*0 = intensity of source [W/m

2]
*θ* = angle between the polarity

of the source and the lens.

Tom Penick [email protected] www.teicontrols.com/notes 1/31/99