1
PHY 212, Constants & Equations Sheet
Fall 2019, Final Exam
General Quadratic Equation: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Pythagorean Theorem: 𝑥 2 + 𝑦 2 = 𝑟 2
1
Triangle area: 𝐴 = 2 𝑏ℎ
Displacement:
Δ𝑥 = 𝑥𝑓 − 𝑥0
Trigonometry:
Kinematics 1Dimensional (𝒙)
Free Fall, No Air Resistance, 1-D
(𝒚) Constant Gravity Accel. Equations:
Position Vector Function:
𝑥
𝑟
tan(𝜃) =
𝑦
𝜃 = sin−1 ( )
𝑟
𝑥
𝜃 = cos −1 ( )
𝑟
𝑣𝑥,𝑎𝑣𝑔 =
𝑣𝑥 =
𝑎𝑥,𝑎𝑣𝑔 =
Δ𝑣𝑥
Δ𝑡
1
𝑣𝑥0 + 𝑣𝑥𝑓
2
𝑣𝑦𝑓 = 𝑣𝑦0 − 𝑔𝑡
𝑣̅𝑦 =
1
𝑟⃗𝑓 (𝑡) = 𝑟⃗0 + 𝑣⃗0 𝑡 + 𝑎⃗𝑡 2
2
© 2019
𝑎𝑥 =
𝑑𝑥
𝑑𝑡
𝑑𝑣𝑥
𝑑2𝑥
=
𝑑𝑡
𝑑𝑡 2
2
2
𝑣𝑥𝑓
= 𝑣𝑥0
+ 2𝑎𝑥 (Δ𝑥)
1
Δ𝑥 = 𝑣𝑥𝑓 𝑡 − 2𝑎𝑥 𝑡 2
Δ𝑥 = 𝑣̅𝑥 𝑡
1
Δ𝑦 = 𝑣𝑦0 𝑡 − 2𝑔𝑡 2
𝑣𝑦0 + 𝑣𝑦𝑓
2
𝑦
𝑥
𝑦
𝜃 = tan−1 ( )
𝑥
Δ𝑥
Δ𝑡
Δ𝑥 = 𝑣𝑥0 𝑡 + 2𝑎𝑥 𝑡 2
𝑣𝑥𝑓 = 𝑣𝑥0 + 𝑎𝑥 𝑡
𝑣̅𝑥 =
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
cos(𝜃) =
Average Acceleration,
Instantaneous Acceleration:
Constant Acceleration Equations:
𝑥=
𝑦
𝑟
sin(𝜃) =
Average Velocity,
Instantaneous Velocity:
Motion 1-Dimensional
(𝒙) Definitions
P.T. Quelet
Quadratic Formula:
2
2
𝑣𝑦𝑓
= 𝑣𝑦0
− 2𝑔(Δ𝑦)
Δ𝑦 = 𝑣̅𝑦 𝑡
Velocity Vector Function:
1
Δ𝑦 = 𝑣𝑦𝑓 𝑡 + 2𝑔𝑡 2
𝑣⃗𝑓 (𝑡) = 𝑣⃗0 + 𝑎⃗𝑡
Last Updated: 9 December 2019
2
Projectile Max
Height:
Projectile Range
(𝚫𝒚 = 𝟎):
Δ𝑦𝑚𝑎𝑥 =
(𝑣0 sin(𝜃0 ))2
2𝑔
𝑣02 sin(2𝜃0 )
Δ𝑥𝑅 =
𝑔
Projectile Range: Δ𝑥 = 𝑣𝑥 𝑡
Newton’s 2nd Law
of Motion:
Static Force of
Friction:
Linear
Resistive
Force:
𝐹𝑓,𝑠 ≤ 𝜇𝑠 𝐹𝑁
Quadratic Air
Drag:
Projectile 𝒙 Accel.
with Air Drag:
𝑊 = 𝐹Δ𝑟 cos(𝜙) = 𝐹⃗ ∙ Δ𝑟⃗
= ∫ 𝐹⃗ ∙ 𝑑𝑟⃗
Hooke’s Law (Spring Force):
Mechanical Energy (ME):
P.T. Quelet
𝐾=
𝑣(𝑡) = 𝑣 𝑇 (1 − 𝑒 −𝑡/𝑇𝒷 )
𝑇𝒷 =
Air Drag Terminal Velocity:
𝑣𝑇 = √
𝜌𝒟𝐴
=−
𝑣 √𝑣 2 + 𝑣𝑦2
2𝑚 𝑥 𝑥
Kinetic Energy (KE):
𝐹𝑓,𝑘 = 𝜇𝑘 𝐹𝑁
Linear Resistive Time Constant:
1
𝐹𝒟 = 𝒟𝜌𝐴𝑣 2
2
𝑎𝑥,𝒟
𝐹𝑔 = 𝑤 = 𝑚𝑔
Gravity Force (Weight):
Linear Falling Velocity:
𝑚𝑔
𝒷
1
𝑚𝑣 2
2
𝐹𝑠 = −𝑘𝑥
𝐸𝑚𝑒𝑐ℎ = 𝐾 + 𝑈
2𝑣0 sin(𝜃0 )
𝑔
𝑣⃗𝑡𝑜𝑡𝑎𝑙 = 𝑣⃗𝑟𝑒𝑙 + 𝑣⃗𝑚𝑒𝑑𝑖𝑢𝑚
Kinetic Force of Friction:
𝐹⃗𝑅 = −𝒷𝑣⃗
𝑣𝑇 =
𝑡𝑅 =
𝑣0 sin(𝜃0 )
𝑔
2
1
𝑥
𝑦(𝑥 ) = 𝑥 tan(𝜃0 ) − 𝑔 (
)
2 𝑣0 cos(𝜃0 )
𝑣⃗𝔸ℂ = 𝑣⃗𝔸𝔹 + 𝑣⃗𝔹ℂ
Σ𝐹⃗ = 𝐹⃗𝑛𝑒𝑡 = 𝑚𝑎⃗
Linear Resistive
Terminal Velocity:
Work:
Projectile Time (𝚫𝒚 = 𝟎):
Projectile Trajectory:
Relative Motion Velocities:
𝑡𝑚𝑎𝑥 =
Max Height Time:
Projectile 𝒚 Accel.
with Air Drag:
Work-KE
Theorem:
2𝑚𝑔
𝒟𝜌𝐴
𝑎𝑦,𝒟
= −𝑔 −
𝜌𝒟𝐴
𝑣 √𝑣 2 + 𝑣𝑦2
2𝑚 𝑦 𝑥
𝑊 = 𝐹𝑛𝑒𝑡 Δ𝑟 cos(𝜃) = 𝐹⃗𝑛𝑒𝑡 ∙ Δ𝑟⃗ = ∫ 𝐹⃗𝑛𝑒𝑡 ∙ 𝑑𝑟⃗
= Σ𝐹Δ𝑟 cos(𝜙) = Δ𝐾
Gravitational Potential Energy:
Elastic (Spring) Potential Energy:
Conservation of Energy:
© 2019
𝑚
𝒷
𝑈𝑔 = 𝑚𝑔𝑦
1
𝑈𝑠 = 𝑘𝑥 2
2
𝐾𝑖 + 𝑈𝑖 + Δ𝐸𝑖𝑛𝑡 = 𝐾𝑓 + 𝑈𝑓
Last Updated: 9 December 2019
3
𝑑𝑈
𝐹𝑥 = −
𝑑𝑥
Conservative Forces, Potential
Functions, 1D & 3D vectors:
Power:
𝑥𝑖
𝑝⃗ = 𝑚𝑣⃗
Kinetic Energy & Momentum:
Conservation of
Momentum:
𝑑𝑝⃗
𝑑𝑣⃗
𝑑𝑚
Σ𝐹⃗ =
=𝑚
= 𝑣⃗
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑚1 − 𝑚2
2𝑚2
) 𝑣1,𝑖 + (
)𝑣
𝑚1 + 𝑚2
𝑚1 + 𝑚2 2,𝑖
1-D Inelastic Collision, Conservation
Equation, Two Discrete Masses:
Centripetal Acceleration:
Angular Displacement:
Δ𝜃 = 𝜃𝑓 − 𝜃0
Rotational Motion
Definitions
𝑥𝐶𝑀 =
𝑚1 𝑥1 + 𝑚2 𝑥2
𝑚1 + 𝑚2
𝑎𝑐 =
𝑣2
= 𝜔2 𝑟
𝑟
P.T. Quelet
Impulse:
𝑡𝑖
Vector Momentum
Conservation:
2𝜋𝑟
𝑣
𝑣2,𝑓 = (
General Center of
Mass:
Σ𝑝⃗𝑖 = Σ𝑝⃗𝑓
2𝑚1
𝑚2 − 𝑚1
) 𝑣1,𝑖 + (
)𝑣
𝑚1 + 𝑚2
𝑚1 + 𝑚2 2,𝑖
𝑟⃗𝐶𝑀 =
1
1
∑ 𝑚𝑖 𝑟⃗𝑖 = ∫ 𝑟⃗𝑖 𝑑𝑚
𝑀
𝑀
𝑖
Centripetal Force:
Average Angular Acceleration,
Instantaneous Angular Acceleration:
𝑇=
𝑡𝑓
𝒥⃗ = Δ(𝑚𝑣⃗) = ∫ 𝐹𝑛𝑒𝑡 𝑑𝑡
𝑚1 𝑣1,𝑖 + 𝑚2 𝑣2,𝑖 = (𝑚1 + 𝑚2 )𝑣𝑓,𝑠𝑦𝑠𝑡𝑒𝑚
Average Angular Velocity,
Instantaneous Angular Velocity:
Rotational to Linear (Tangential) Relations:
Angular Period with
Linear Velocity:
𝑝2
𝐾=
2𝑚
𝑚1 𝑣1,𝑖 + 𝑚2 𝑣2,𝑖 = 𝑚1 𝑣1,𝑓 + 𝑚2 𝑣2,𝑓
1
1
1
1
2
2
2
2
𝑚1 𝑣1,𝑖
+ 𝑚2 𝑣2,𝑖
= 𝑚1 𝑣1,𝑓
+ 𝑚2 𝑣2,𝑓
2
2
2
2
1-D Elastic Collision, Conservation
Equations, Two Discrete Masses:
1-D Center of Mass,
Two Discrete Masses:
𝑊 = 𝑃Δ𝑡 = ∫ 𝑃 𝑑𝑡
𝑡𝑖
𝒥⃗ = Δ(𝑚𝑣⃗) = 0 → (𝑚𝑣⃗ )𝑖 = (𝑚𝑣⃗)𝑓
𝑣1,𝑓 = (
𝑟⃗𝑖
𝑡𝑓
Power and Energy:
Generalized Version of Newton’s
2nd Law of Motion:
𝑟⃗𝑓
Δ𝑈 = − ∫ 𝐹𝑥 𝑑𝑥 = − ∫ 𝐹⃗ ∙ 𝑑𝑟⃗
𝑊 𝑑𝑊
𝑃=
=
= 𝐹𝑣
Δ𝑡
𝑑𝑡
Momentum:
Final
Velocities, 1-D
Elastic Coll:
𝑥𝑓
⃗⃗𝑈
𝐹⃗ = −∇
𝜔𝑎𝑣𝑔 =
𝛼𝑎𝑣𝑔 =
𝑠 = Δ𝜃 𝑟
Angular Period with
Angular Velocity:
© 2019
𝐹𝑐 =
𝑚𝑣 2
= 𝑚𝜔2 𝑟
𝑟
Δ𝜃
Δ𝑡
Δ𝜔
Δ𝑡
𝜔=
𝛼=
𝑑𝜃
𝑑𝑡
𝑑𝜔
𝑑2𝜃
=
𝑑𝑡
𝑑𝑡 2
𝑣t = 𝜔𝑟
𝑎𝑡 = 𝛼𝑟
𝑇=
2𝜋
𝜔
Last Updated: 9 December 2019
4
Rotational
Kinematics (𝜽)
Constant Angular
Acceleration Equations:
𝜔
̅=
𝜔0 + 𝜔𝑓
2
Moment of Inertia,
Point Masses:
I = Σ𝑚𝑟 2
Moment of Inertia,
Parallel Axis
Theorem:
I = ICM + 𝑚𝑑 2
Moment of Inertia,
Extended Objects:
Torque:
Rotational Kinetic Energy:
1 2
I𝜔
2
I𝑖 𝜔𝑖 = I𝑓 𝜔𝑓
Escape Velocity:
P.T. Quelet
𝑀1 𝑚2
𝑟̂
𝑟2
𝐺𝑀
𝑅2
𝑣𝑜𝑟𝑏𝑖𝑡
2
ISolid Sphere = 𝑀𝑅2
5
Σ𝜏⃗ = 𝜏⃗𝑛𝑒𝑡 = I𝛼⃗ = 𝑟⃗ × 𝐹⃗𝑛𝑒𝑡 =
2𝐺𝑀
𝑣𝑒𝑠𝑐 = √
𝑅
⃗⃗
𝑑𝕃
𝑑𝑡
𝑊 = Σ𝜏Δ𝜃
= Δ𝐾𝑟𝑜𝑡
Angular Momentum Vector:
Vector Angular Momentum
Conservation:
𝐺 ≡ 6.67 × 10−11
Gravitation Constant:
𝑈𝑔,𝑟 = −𝐺
𝑇2 = (
Kepler’s 3rd Law:
Maximum Height 𝒓𝒎𝒂𝒙
Launched Gravit. Object:
© 2019
⃗𝕃⃗ = 𝑟⃗ × 𝑝⃗
⃗⃗ = 0 → Σ𝕃
⃗⃗𝑖 = Σ𝕃
⃗⃗𝑓
Δ𝕃
Radially Varying Gravitational
Potential Energy:
𝐺𝑀
=√
𝑟
Iz = Ix + Iy
1
1
2
𝐾𝑟𝑜𝑙𝑙 = 𝐾𝐶𝑀 + 𝐾𝑟𝑜𝑡 = 𝑚𝑣𝐶𝑀
+ ICM 𝜔2
2
2
Conservation of
Angular Momentum:
Orbit Velocity:
I = ∫ 𝑟 2 𝑑𝑚 = ∫ 𝑟 2 𝜌𝑑𝑉
Rotational Work KE Theorem:
𝕃 = 𝑚𝑣𝑟 sin(𝜙) = I𝜔
𝑔=
1
1
𝑀𝑅2
2
Net Torque:
Angular Momentum:
Local Gravitational
Acceleration:
Δ𝜃 = 𝜔𝑓 𝑡 − 2𝛼𝑡 2
IDisc =
𝜏⃗ = 𝑟⃗ × 𝐹⃗
𝐾𝑟𝑜𝑡 =
𝐹⃗𝑔 = 𝐺
Δ𝜃 = 𝜔
̅𝑡
Mom. of Inertia, Planar Object,
Perpendicular Axis Theorem:
Rolling Kinetic Energy (No slipping):
Universal
Gravitation:
𝜔𝑓2 = 𝜔02 + 2𝛼(Δ𝜃)
Moment of Inertia, Continuous
Mass Distribution:
IHoop = 𝑀𝑅2
𝜏 = 𝑟𝐹 sin(𝜙)
1
Δ𝜃 = 𝜔0 𝑡 + 2𝛼𝑡 2
𝜔𝑓 = 𝜔0 + 𝛼𝑡
𝑁 ∙ 𝑚2
𝑘𝑔2
𝑀1 𝑚2
𝑟
4𝜋 2 3
)𝑎
𝐺𝑀
1
1
𝑣𝑖 = √2𝐺𝑀 ( −
)
𝑅 𝑟𝑚𝑎𝑥
Last Updated: 9 December 2019
5
Position Function, Simple
Harmonic Motion (SHM):
Velocity
Function, SHM:
Acceleration
Function, SHM:
𝑣𝑥 (𝑡) = −𝓌𝐴0 sin(𝓌𝑡 + 𝜑)
Frequency Definition:
Angular Frequency, 1-D Ideal
Spring and Mass:
𝑥(𝑡) = 𝐴0 cos(𝓌𝑡 + 𝜑)
𝑓=
1 𝓌
=
𝑇 2𝜋
𝑎𝑥 (𝑡) = −𝓌 2 𝐴0 cos(𝓌𝑡 + 𝜑)
𝑇=
Period Definition:
Maximum Speed,
SHM:
𝑣𝑥,𝑚𝑎𝑥 = 𝓌𝐴0
Maximum Acceleration,
Magnitude SHM:
SHM Velocity,
Function of Position:
𝑣𝑥 (𝑥 ) = ±𝓌√𝐴20 − 𝑥 2
Total Energy in Simple
Harmonic Motion:
Kinetic Energy,
SHM:
1
𝑚𝓌 2 𝐴20 sin2 (𝓌𝑡 + 𝜑)
2
𝐾=
Angular Frequency, Simple
Pendulum:
Angular Frequency,
Physical Pendulum:
𝓌=√
Elastic Potential
Energy, SHM:
𝑔
ℓ
𝑚𝑔𝑑
𝓌=√
I
SHM Damped Oscillation Position and
Angular Frequency:
𝑥 (𝑡 ) =
−𝒷
𝐴0 𝑒 2𝑚
1D Linear Traveling
Wave, General Form:
1D Linear Traveling
Wave, Phase Speed:
Angular Frequency:
P.T. Quelet
𝑦(𝑥, 𝑡) = 𝐴0 sin(𝑥 − 𝑣𝑡)
𝑣 = 𝜆𝑓 =
𝜆 𝓌
=
𝑇 𝓀
1 2𝜋
=
𝑓 𝓌
𝑎𝑥,𝑚𝑎𝑥 = 𝓌 2 𝐴0
𝐸𝑆𝐻𝑀,𝑡𝑜𝑡𝑎𝑙 =
𝑈𝐸 =
1 2
𝑘𝐴 cos 2 (𝓌𝑡 + 𝜑)
2 0
ℓ
𝑇 = 2𝜋√
𝑔
Period, Simple
Pendulum:
I
𝑇 = 2𝜋√
𝑚𝑔𝑑
cos(𝓌𝒷 𝑡 + 𝜑)
𝑘
𝒷 2
√
𝓌𝒷 =
−( )
𝑚
2𝑚
𝐹⁄
𝑚
√(𝓌𝐹2 − 𝜔 2 )2 − (𝒷𝓌𝐹 )
𝑚
1D Linear Traveling
Wave, Angular Form:
Angular Wavenumber:
© 2019
2
𝑦(𝑥, 𝑡) = 𝐴0 sin(𝓀𝑥 − 𝓌𝑡)
1D Linear Traveling Wave,
Partial Differential Equation:
𝓌 = 2𝜋𝑓
1 2
𝑘𝐴
2 0
Period, Simple
Pendulum:
𝐴=
SHM Amplitude of General Forced (ang. freq. 𝝎𝑭 ) or
Damped (𝓫) Oscillation:
𝑘
𝓌=√
𝑚
𝜕2𝑦
𝜕2𝑦
2
=
𝑣
𝜕𝑡 2
𝜕𝑥 2
𝓀=
2𝜋
𝜆
Last Updated: 9 December 2019
6
Transverse Velocity:
𝑣𝔱 =
𝜕𝑦
𝜕𝑡
Power Transmitted by
Traveling Wave:
1
𝑃 = μ𝓌 2 𝐴20 𝑣
2
Intensity Definition:
Common Log Intensity
Scale (Decibels):
ℐ
𝛽 = 10 log ( )
ℐ0
Spherical Wave
Intensity:
Hearing Sound Intensity
Threshold Constant:
Doppler Shifted
Frequency:
ℐ0 = 10−12
𝑓𝐷 = 𝑓 (
𝐹𝑇
𝑣=√
μ
Traveling Wave Phase Speed,
String Under Tension:
W
m2
ℐ=
Sound Speed in Air
Approximation:
𝑣 + 𝑣𝑜𝑏𝑠
)
𝑣 − 𝑣𝑠𝑜𝑢𝑟𝑐𝑒
ℐ=
Superposition, Incident and Reflected Identical
Traveling Waves, Standing Wave:
𝑣𝑠 [m⁄s] = 331 + 0.6 𝑇[°𝐶]
𝑦+𝑣 + 𝑦−𝑣 = (2𝐴0 sin(𝓀𝑥)) cos(𝓌𝑡)
𝑓𝑛 = 𝑛𝑓1
(𝑛 ∈ ℤ + )
Frequency, Standing Wave on
String (Fixed Both Ends):
Wavelength of Standing
Waves, Fixed on Both Ends, or
Open on Both Ends:
2ℓ
𝜆𝑛 =
𝑛
Frequency of Standing Waves,
Fixed on Both Ends, or Open on
Both Ends:
Wavelength of Standing
Waves, Fixed on One End,
Open on Other:
4ℓ
𝑛
(𝑛 ∈ odd+ )
Frequency of Standing Waves,
Fixed on One End, Open on
Other:
Sea Level Density:
Pressure at Depth in Fluid:
P.T. Quelet
𝜑
Δ𝑟
=
2𝜋
𝜆
Phase Shift, Path Difference,
Wave Interference
Overtones, Fundamental to
Harmonics Relation:
Density Definition:
𝑃
4𝜋𝑟 2
𝜑
𝜑
𝑦+𝑣 (𝑥, 𝑡) + 𝑦+𝑣 (𝑥, 𝑡, 𝜑) = (2𝐴0 cos ( )) sin (𝓀𝑥 − 𝓌𝑡 + )
2
2
Superposition, Two Identical Traveling
Waves, Phase Difference 𝝋:
Average Frequency:
𝑃
𝐴
𝜆𝑛 =
𝑓𝑎𝑣𝑔 =
𝑓1 + 𝑓2
2
𝜌=
Beats Frequency:
𝑚
𝑉
Pressure Definition:
𝜌0 ≈ 1.29 kg⁄m3
𝓅 = 𝜌𝑔ℎ
© 2019
𝑓𝑛 =
𝑛 𝐹𝑇
√
2ℓ μ
𝑓𝑛 =
𝑛
𝑣
2ℓ
𝑛
𝑣
4ℓ
(𝑛 ∈ odd+ )
𝑓𝑛 =
𝑓𝔟 = |𝑓1 − 𝑓2 |
𝓅=
𝐹
𝐴
Sea Level Pressure:
𝓅0 ≈ 1.013 × 105 Pa
Total Pressure at
Depth in Liquid:
𝓅𝑡𝑜𝑡𝑎𝑙 = 𝓅0 + 𝜌𝑔ℎ
Last Updated: 9 December 2019
7
𝐹1 𝐹2
=
𝐴1 𝐴2
Pascal’s Principle:
𝐹𝑏 = 𝑤 − 𝑤𝑖𝑛 𝑓𝑙𝑢𝑖𝑑
Buoyancy Force:
Fraction Submerged
(Floating):
Volume Flow
Rate:
Bernoulli’s
Equation:
𝑓𝑟𝑎𝑐. =
ℱ=
Archimedes’ Principle:
𝜌𝑜𝑏𝑗𝑒𝑐𝑡
𝜌𝑓𝑙𝑢𝑖𝑑
𝑑𝑉
= 𝐴𝑐𝑠 𝑣
𝑑𝑡
𝑣22 = 𝑣12 + 2𝑔ℎ
Mass of Proton:
𝑚 +p = 1.67 × 10−27 𝑘𝑔
𝐹𝐸 = 𝑘𝐸
𝐸⃗⃗ =
Linear Charge
Density:
𝐸⃗⃗ = 𝑘𝑒
𝜆=
Electric Dipole Moment:
Electric Field Approx.,
Dipole on Axis:
P.T. Quelet
𝑞
ℓ
𝑞
𝑟̂
𝑟2
𝐹⃗𝐸
𝑞0
𝐸⃗⃗ ≈ 𝑘𝑒
2𝔭
⃗⃗
𝑟3
𝓅1 − 𝓅2
𝜌𝑔
𝑘𝐸 = 9.00 × 109 𝑁𝑚2 ⁄𝐶 2
𝒌𝑬 to 𝝐𝟎 Equivalency:
Electrostatic Force
Vector:
𝑘𝐸 =
Electric Field, Continuous
Charge Distribution:
𝜎=
𝑞
𝐴
Torque on Dipole in
Electric Field:
Electric Field of Dipole,
Perp. Bisect Plane:
© 2019
1
4𝜋𝜖0
𝑞1 𝑞2
𝑟̂
𝑟 2 1,2
𝐹⃗𝐸 = 𝑘𝑒
𝐹⃗𝐸 = 𝑞𝐸⃗⃗
Electric Force Vector Definition:
Surface Charge
Density:
𝔭⃗⃗ = 𝑞𝑠⃗
1
1
𝓅1 + 𝜌𝑣12 = 𝓅2 + 𝜌𝑣22
2
2
𝑚 −e = 9.11 × 10−31 𝑘𝑔
Coulomb’s Constant:
|𝑞1 ||𝑞2 |
𝑟2
𝑑𝑚
𝑑𝑡
𝐴1 𝑣1 = 𝐴2 𝑣2
𝑦𝑚𝑎𝑥 =
Mass of Electron:
𝜖0 = 8.85 × 10−12 𝐶 2 ⁄𝑁𝑚2
Electric Field Vector Definition:
Electric Field, Point
Charge:
Bernoulli’s
Principle:
Maximum Fluid Height:
𝑞𝑡𝑜𝑡𝑎𝑙 = 𝑛 𝑞p+ 𝑜𝑟 e−
Electrostatic Force
Magnitude (Coulomb’s Law):
𝑚̇ =
Conservation of Flow Rate
(Incompressible):
Torricelli Flow:
Permittivity of Free
Space:
𝐹𝑏 = 𝜌𝑓𝑙𝑢𝑖𝑑 𝑔𝑉𝑜𝑏𝑗𝑒𝑐𝑡 = 𝑤𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝
Mass Flow Rate:
1
1
𝓅1 + 𝜌𝑣12 + 𝜌𝑔ℎ1 = 𝓅2 + 𝜌𝑣22 + 𝜌𝑔ℎ2
2
2
Total Charge:
𝓅ℊ = 𝓅𝑎𝑏𝑠 + 𝓅0
Gage Pressure:
𝐸⃗⃗ = 𝑘𝑒 ∫
𝑑𝑞
𝑟̂
𝑟2
Volume Charge Density:
𝜚=
𝑞
𝑉
𝜏⃗ = 𝔭⃗⃗ × 𝐸⃗⃗
𝐸⃗⃗ ≈ −𝑘𝑒
𝔭⃗⃗
𝑟3
Last Updated: 9 December 2019
8
Electric Field, Infinite
Line of Charge:
𝐸⃗⃗ = 𝑘𝑒
2𝜆
𝑖̂
𝑟
E-Field, Finite Line of
Charge, @Midpoint
𝐸⃗⃗ =
Electric Field, Charged Ring in xy Plane Centered @ Origin,
Perpendicular Axis above Center:
Electric Flux
Definition:
𝐸⃗⃗ =
𝜎
𝑟̂
2𝜖0
𝑥√𝑥 2 + (𝐿/2)2
𝐸⃗⃗ = 𝑘𝑒 𝑄
Electric Field, Charged Disk in xy Plane Centered @ Origin,
Perpendicular Axis above Center:
Electric Field, Infinite Plane
of Charge:
𝑘𝑒 𝑄
𝐸⃗⃗ =
𝑧
𝑘̂
(𝑧 2 + 𝑅2 )3/2
𝜎
𝑧
[1 −
] 𝑘̂
2𝜀0
√𝑧 2 + 𝑅2
Electric Field, Conductor or
Parallel Plate Capacitor:
𝐸⃗⃗ =
Electric Flux, Continuous
Charge Distribution:
Φ𝐸 = 𝐸⃗⃗ ∙ 𝐴⃗ = 𝐸𝐴 cos(𝜃)
𝜎
𝑞𝑡𝑜𝑡𝑎𝑙
𝑟̂ =
𝑟̂
𝜖0
𝐴𝜖0
Φ𝐸 = ∫ 𝐸⃗⃗ ∙ 𝑑𝐴⃗
𝑆
Φ𝐸 = ∯ 𝐸⃗⃗ ∙ 𝑑𝐴⃗ =
Gauss’ Law for Electric Fields, Electric Flux Form:
𝑖̂
𝑆
𝑞𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
𝜖0
B
Electric Potential
Energy Difference:
𝛥𝑈𝑉 = −𝑞0 ∫ 𝐸⃗⃗ ∙ 𝑑𝑠⃗
Electric Potential Energy Difference,
Uniform Electric Field:
𝛥𝑈𝑉 = −𝑞0 𝐸𝑑
A
B
𝛥𝑈𝑉
Δ𝑉 =
= − ∫ 𝐸⃗⃗ ∙ 𝑑𝑠⃗
𝑞0
Electric Potential
Difference (Voltage):
Electric Potential (Voltage)
Difference, Uniform Electric Field:
𝛥𝑉
= −𝐸𝑑
A
𝑉 = 𝑘𝑒
Voltage, Point Charge:
Electric Field from
Voltage, 1D:
𝐸𝑥 = −
Capacitance Definition:
𝑑𝑉
𝑑𝑥
𝑞
𝑟
Voltage, Continuous
Charge Distribution:
Electric Field Vector, Voltage
Potential Function, 3D:
𝐶=
𝑞
Δ𝑉
𝑉 = 𝑘𝑒 ∫
⃗⃗𝑉 = − (
𝐸⃗⃗ = −∇
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑖̂ +
𝑗̂ +
𝑘̂)
𝜕𝑥
𝜕𝑦
𝜕𝑧
Parallel Plate Capacitance:
𝐶 = 𝜖0
1
Equivalent Capacitance,
Parallel Config.:
𝐶𝑒𝑞,𝑝 = 𝐶1 + 𝐶2 + ⋯
+ 𝐶𝑁
Capacitor Stored Energy:
𝑈𝐶 =
P.T. Quelet
Equivalent Capacitance,
Series Config.:
𝑞2 1
1
= 𝑞Δ𝑉 = 𝐶 (Δ𝑉 )2
2𝐶 2
2
© 2019
𝑑𝑞
𝑟
𝐶𝑒𝑞,𝑠
=
𝐴
𝑑
1
1
+ +⋯
𝐶1 𝐶2
1
+
𝐶𝑁
Capacitance with Dielectric:
𝐶𝜅 = 𝜅𝐶
Last Updated: 9 December 2019
9
Electric Current, Fundamental Charge
Carriers ( −e):
Average Electric Current:
𝐼=
Electric Current Density:
𝐽=
Resistivity / Conductivity
Definition:
Electric Resistor
Power Transfer:
Conduction-electron
Number Density:
𝐼 = 𝓃 −e𝑣𝑑 𝐴
Δ𝑞
Δ𝑡
1
𝜎𝑐
𝑃 = 𝐼Δ𝑉 = 𝐼 2 ℛ =
𝐼=
Instantaneous Electric Current:
𝐼
= 𝓃𝑞e 𝑣𝑑 = 𝜎𝑐 𝐸
𝐴
𝜌ℛ =
Ohm’s Law:
𝐼=
Δ𝑉
ℛ
Kirchhoff’s Junction Rule:
∑𝐼 = 0
jct
Equivalent Resistance,
Parallel Config.:
Real Battery Maximum
Current:
ℓ
𝐴
1
1
+
+⋯
ℛ1 ℛ 2
1
+
ℛ𝑁
𝐼𝑏𝑎𝑡𝑡 =
Kirchhoff’s Loop Rule:
ℰ
ℛ𝑏𝑎𝑡𝑡
∑ Δ𝑉 = 0
loop
𝑞(𝑡) = 𝑞𝑚𝑎𝑥 (1 − 𝑒 −𝑡 ⁄ℛ𝐶 ) = 𝐶ℰ(1 − 𝑒 −𝑡 ⁄ℛ𝐶 )
𝐼(𝑡) = 𝐼𝑚𝑎𝑥 𝑒 −𝑡 ⁄ℛ𝐶 =
RC Circuit, Current during Charging:
ℰ −𝑡 ⁄ℛ𝐶
𝑒
ℛ
𝑞(𝑡) = 𝑞𝑚𝑎𝑥 𝑒 −𝑡 ⁄ℛ𝐶 = 𝐶𝑉𝑚𝑎𝑥 𝑒 −𝑡⁄ℛ𝐶
RC Circuit, Charge on Discharging Capacitor:
𝐼 (𝑡) = −𝐼𝑚𝑎𝑥 𝑒 −𝑡 ⁄ℛ𝐶 = −
RC Circuit, Current during Discharging:
Magnetic Force Vector,
Moving Charge:
𝐹𝐵 = 𝑞𝑣𝐵 sin 𝜙
Radius, Angular Frequency, and Period of Charge,
Uniform Magnetic Field:
© 2019
𝑉𝑚𝑎𝑥 −𝑡 ⁄ℛ𝐶
𝑒
ℛ
⃗⃗ )
𝐹⃗𝐵 = 𝑞(𝑣⃗ × 𝐵
⃗⃗)
𝐹⃗𝐸𝑀 = 𝑞𝐸⃗⃗ + 𝑞(𝑣⃗ × 𝐵
Total Electromagnetic Force Vector on Charge 𝒒:
P.T. Quelet
ℛ𝑒𝑞,𝑝
=
jct
RC Circuit, Charge on Charging Capacitor:
Magnetic (Lorenz) Force,
Moving Charge:
Δ𝑉 = 𝐼ℛ
𝐸𝑘𝑊∗ℎ𝑟
≡ 3.60 × 106 J
∑ 𝐼in = ∑ 𝐼out
jct
𝑑𝑞
𝑑𝑡
Kilowatt-hour Definition:
Δ𝑉𝑏𝑎𝑡𝑡 = ℰ − 𝐼ℛ𝑏𝑎𝑡𝑡
Real Battery Terminal Voltage:
𝑉
ℛ = 𝜌ℛ
(Δ𝑉 )2
ℛ
ℛ𝑒𝑞,𝑠 = ℛ1 + ℛ2 + ⋯
+ ℛ𝑁
𝑁 −e
Cylindrical Wire Resistance:
1
Equivalent Resistance,
Series Config.:
𝓃=
𝑟=
𝑚𝑣
𝑞𝐵
𝜔=
𝑞𝐵
𝑚
𝑇=
2𝜋𝑚
𝑞𝐵
Last Updated: 9 December 2019
10
Velocity Selector:
𝑣=
𝐸
𝐵
Mass
Spectrometer:
Magnetic (Lorenz) Force,
Current-Carrying Wire:
𝐹𝐵 = 𝐼ℓ𝐵 sin 𝜙
Torque on Current
Windings:
𝜏 = 𝑁𝐼𝐴𝐵 sin 𝜙
Magnetic Dipole Moment
Vector:
𝜇⃗𝐵 = 𝑁𝐼𝐴⃗
Magnetic Field
Differential from Moving
Charge
(Biot-Savart Law):
Ampere’s Law, for
Magnetic Fields:
⃗⃗ =
𝑑𝐵
𝑟=
Torque Vector on
Current Windings:
𝐹𝐵 𝜇0 𝐼1 𝐼2
=
ℓ
2𝜋𝑎
⃗⃗ ∙ 𝐴⃗ = 𝐵𝐴 cos(𝜃)
Φ𝐵 = 𝐵
Faraday’s Law for Induction:
Motional EMF:
ℰ𝑀 = 𝐵ℓ𝑣
Rotating Electric
Generator Magnetic Flux:
P.T. Quelet
Ideal
Transformers:
Φ𝐵 = 𝐵𝐴 cos(𝓌𝑡)
⃗⃗ =
𝑑𝐵
Magnetic Field, Solenoid
( 𝓵 ≫ 𝒓) :
⃗⃗ =
𝐵
𝑆
=0
𝐵𝑙𝑜𝑜𝑝 =
𝐵𝑆 =
𝜇0 𝐼
2𝑟
𝜇0 𝑁𝐼
= 𝜇0 𝔫𝐼
ℓ
⃗⃗ ∙ 𝑑𝐴⃗
Φ𝐵 = ∫ 𝐵
𝑆
𝑑Φ𝐵
𝑑
= − (𝑁𝐵𝐴 cos(𝜃)) = ∮ 𝐸⃗⃗ ∙ 𝑑𝑠⃗
𝑑𝑡
𝑑𝑡
𝑉𝑝
𝑉𝑠
=
𝑁𝑝 𝑁𝑠
Ideal Transformers:
Rotating Electric Generator
EMF:
© 2019
𝜇0 𝐼(𝑑𝑠⃗ × 𝑟̂ )
4𝜋
𝑟2
𝜇0 𝐼𝑅2
𝑘̂
2(𝑧 2 + 𝑅2 )3/2
Magnetic Flux,
Continuous Field
Distribution:
ℰ = −𝑁
Tm
A
⃗⃗ ∙ 𝑑𝐴⃗
Φ𝐵 = ∯ 𝐵
Magnetic Field, center of
Single Current Loop:
Magnetic Field, xy Plane Ring @ Origin,
Perpendicular Axis Charge above Center:
Magnetic Flux
Definition:
𝜇0 ≡ 4𝜋 × 10−7
Gauss’ Law for Magnetic Fields,
Magnetic Flux Form:
𝜇0 𝐼
2𝜋𝑟
𝑞 2 𝐵2 𝑟 2
2𝑚
⃗⃗) = 𝜇⃗𝐵 × 𝐵
⃗⃗
𝐹⃗𝐵 = 𝐼(𝐴⃗ × 𝐵
Magnetic Field Differential
from Current
(Biot-Savart Law):
𝜇0 𝑞(𝑣⃗ × 𝑟⃗)
4𝜋
𝑟3
𝐾𝐶 =
⃗⃗)
𝐹⃗𝐵 = 𝐼(ℓ⃗⃗ × 𝐵
Permeability of Free Space
(Magnetism):
𝐵𝑤𝑖𝑟𝑒 =
Magnetic Force per Length,
Long Current-Carrying Wires:
Cyclotron Kinetic
Energy:
Magnetic Force Vector,
Current-Carrying Wire:
⃗⃗ ∙ 𝑑𝑠⃗ = 𝜇0 𝐼𝑝𝑒𝑛𝑒𝑡𝑟𝑎𝑡𝑒
∮𝐵
Magnetic Field, 𝒓 from CurrentCarrying Long Straight Wire:
𝐸 𝑚
( )
𝐵0 𝐵 𝑞
𝑉𝑝 𝐼𝑝 = 𝑉𝑠 𝐼𝑠
ℰ = 𝑁𝐵𝐴𝓌 sin(𝓌𝑡)
Last Updated: 9 December 2019
11
Φ𝐵 ≡ 𝐿𝐼
Inductance Defined:
Inductor EMF:
𝜇0 𝑁 2 𝐴
𝐿𝑆 =
ℓ
Inductance, Solenoid
( 𝓵 ≫ 𝒓) :
𝐼 (𝑡) = 𝐼𝑚𝑎𝑥 (1 − 𝑒 −𝑡⁄𝒯ℛ𝐿 ) =
Charge on Capacitor,
LC Oscillating Circuit:
Energy Density,
Electric Field:
𝐼(𝑡) = 𝐼𝑚𝑎𝑥 𝑒 −𝑡 ⁄𝒯ℛ𝐿 =
𝑞(𝑡) = 𝑞0 cos(𝓌𝑡 + 𝜑)
1
𝜖 𝐸2
2 0
𝑢𝐸 =
ℇ −𝑡 ⁄𝒯
ℛ𝐿
𝑒
𝑅
AC Current,
Capacitor:
𝒾𝑅 (𝑡) =
𝒾𝐿 (𝑡) =
RC Filter, AC,
Crossover Freq.:
Series RLC
Current:
AC Series RLC
Resonance Freq.:
Root-Mean-Square
Volt. & Current:
𝐼=
ℰ0
𝒵 = √𝑅2 + (𝛸𝐿 − 𝛸𝐶 )2
𝒵
𝜔0 =
1
√𝐿𝐶
𝐼𝑅𝑀𝑆 =
𝐼
√2
𝑓0 =
1
2𝜋√𝐿𝐶
𝑉𝑅𝑀𝑆 =
𝑉
√2
1
√𝐿𝐶
𝐵2
2𝜇0
AC Resistor Voltage:
𝑉𝑅 = ℰ0 cos(𝜙)
1
𝜔𝐶
Capacitive Reactance:
Χ𝐶 =
Inductive Reactance:
Χ𝐿 = 𝜔𝐿
Series RLC Voltages:
𝒾(𝑡) = 𝒾𝑅 = 𝒾𝐿 = 𝒾𝐶 = 𝐼 cos(𝜔𝑡 + 𝜙)
RLC Max Current &
Impedance:
P.T. Quelet
𝑉𝑅
cos(𝜔𝑡) = 𝐼𝑅 cos(𝜔𝑡)
𝑅
1
𝑅𝐶
𝐿
ℛ
ℰ (𝑡) = ℰ0 cos(𝜔𝑡 + 𝜙)
𝑉𝐶
𝜋
𝜋
cos (𝜔𝑡 − ) = 𝐼𝐿 cos (𝜔𝑡 − )
𝜔𝐿
2
2
𝜔𝑐 =
𝓌𝐿𝐶 =
𝑢𝐵 =
𝜋
𝜋
𝒾𝐶 (𝑡) = 𝜔𝐶𝑉𝐶 cos (𝜔𝑡 + ) = 𝐼𝐶 cos (𝜔𝑡 + )
2
2
AC Current,
Inductor:
𝒯ℛ𝐿 =
Angular (Resonant) Frequency,
LC Oscillating Circuit:
Ideal AC EMF (Generator, starting from maximum):
AC Resistor Current:
ℇ
(1 − 𝑒 −𝑡 ⁄𝒯ℛ𝐿 )
𝑅
RL Circuit, e-folding
Time Constant:
Energy Density,
Magnetic Field:
𝑑𝐼
𝑑𝑡
1
𝑈𝐿 = 𝐿𝐼 2
2
Potential Energy Stored in an
Inductor:
RL Circuit, Current with an Energizing Inductor:
RL Circuit, Current with a
De-energizing Inductor:
ℰ𝐿 = −𝐿
ℰ (𝑡) = 𝓋𝑅 (𝑡) + 𝓋𝐿 (𝑡) + 𝓋𝐶 (𝑡)
AC RLC Max
Voltages:
RLC Phase:
ℰ02 = 𝑉𝑅2 + (𝑉𝐿 − 𝑉𝐶 )2
tan(𝜙) =
𝛸𝐿 − 𝛸𝐶 𝑉𝐿 − 𝑉𝐶
=
𝑅
𝑉𝑅
ℰ0
cos(𝜔𝑡 + 𝜙)
𝒵
AC RLC Current:
𝒾𝑅𝐿𝐶 (𝑡) =
AC Res. Power:
2
𝑃𝑅,𝐴𝐶 = 𝐼𝑅𝑀𝑆 ℰ𝑅𝑀𝑆 = 𝐼𝑅𝑀𝑆
𝑅
© 2019
Last Updated: 9 December 2019
12
AC Capacitor
Power:
1
𝑃𝐶,𝐴𝐶 (𝑡) = − 𝜔𝐶𝑉𝐶2 sin(2𝜔𝑡)
2
Displacement
Current:
𝐼𝑑 = 𝜖0
𝑑Φ𝐸
𝑑𝑡
Maxwell’s Equations,
Differential Form:
Speed of Light
(in vacuum):
𝑐=
1
√𝜖0 𝜇0
Traveling E.M.
Radiation Speed:
Poynting
Vector:
Ampere-Maxwell Law, for
circulating Magnetic Fields:
⃗⃗ ∙ 𝐸⃗⃗ =
∇
=
𝜚
𝜖0
𝑆⃗ =
⃗⃗ ∙ 𝐵
⃗⃗ = 0
∇
𝐸
m
≈ 3.0 × 108
𝐵
s
𝑐 = 𝜆𝑓 =
1
⃗⃗
𝐸⃗⃗ × 𝐵
𝜇0
𝓌
𝓀
P.T. Quelet
𝓅𝑎𝑏𝑠 =
⃗⃗ ∙ 𝑑𝑠⃗ = 𝜇0 (𝐼𝑝𝑒𝑛𝑒𝑡𝑟𝑎𝑡𝑒 + 𝜖0
∮𝐵
⃗⃗ × 𝐸⃗⃗ = −
∇
⃗⃗
𝜕𝐵
𝜕𝑡
E.M. Radiation Doppler Shift
(Non-Relativistic):
𝑢𝑎𝑣𝑔 =
𝑑Φ𝐸
)
𝑑𝑡
⃗⃗ × 𝐵
⃗⃗ = 𝜇0 (𝐽⃗ + 𝜖0
∇
Vector Direction of Electromagnetic
(E.M.) Radiation:
Average Energy Density,
E.M. Wave:
𝜕𝐸⃗⃗
)
𝜕𝑡
⃗⃗ = 𝑐⃗
𝐸⃗⃗ × 𝐵
𝑓𝐷,𝐸𝑀 = √
𝑐+𝑣
𝑐−𝑣
2
1
𝐵𝑚𝑎𝑥
2
𝜖0 𝐸𝑚𝑎𝑥
=
2
2𝜇0
2
2
𝑃
𝐸𝑚𝑎𝑥
𝑐𝐵𝑚𝑎𝑥
ℐ = = 𝑆𝑎𝑣𝑔 = 𝑐𝑢𝑎𝑣𝑔 =
=
𝐴
2𝜇0 𝑐
2𝜇0
Electromagnetic Radiation
Intensity:
Absorption & Reflection
Radiation Pressure:
𝑃ℰ,𝐴𝐶 = 𝐼𝑅𝑀𝑆 ℰ𝑅𝑀𝑆 cos(𝜙)
AC EMF Power:
ℐ 𝑆𝑎𝑣𝑔
=
𝑐
𝑐
𝓅𝑟𝑒𝑓 = 2𝓅𝑎𝑏𝑠
© 2019
Polarization
(Malus’ Law):
ℐ(𝜃) = ℐ0 cos 2 (𝜃)
Last Updated: 9 December 2019