Electromagnetism Formulas: The Hidden Force Behind Your Phone | ORBITECH

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Electrostatics: The Invisible Push and Pull

Formulas that describe how stationary charges create electric fields and forces

Coulomb's Law law

F = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{r^{2}}

Formes alternatives

$F = k_{e} \frac{q_{1} q_{2}}{r^{2}}$ — where $k_{e}$ = 1/(4π $ε$ _0) = 8.988 × 10^9 N·m²/C²

Symbole	Signification	Unité
`F`	electrostatic force Attractive if charges opposite, repulsive if same sign	N
`q_1, q_2`	point charges Elementary charge e = 1.602 × 10^{-19} C	C
`r`	distance between charges Measured center-to-center	m
`\varepsilon_0`	vacuum permittivity 8.854 × 10^{-12} F/m

Dimensions : $[M] [L] [T^{- 2}]$

Exemple : Two 1 nC charges separated by 5 cm in Auckland feel a force of 3.6 × 10^{-6} N

Electric Field of a Point Charge law

E = \frac{1}{4 π ε_{0}} \frac{q}{r^{2}}

Symbole	Signification	Unité
`E`	electric field strength Also V/m	N/C
`q`	source charge Can be positive or negative	C
`r`	distance from charge Radial distance	m

Dimensions : $[M] [L] [T^{- 3}] {[I]}^{- 1}$

Exemple : A 2 nC charge creates E = 3.6 × 10^3 N/C at 10 cm away

Gauss's Law law

\oint_{S} 𝐄 \cdot d 𝐀 = \frac{Q_{enc}}{ε_{0}}

Symbole	Signification	Unité
`E`	electric field Integrated over closed surface S	N/C
`dA`	infinitesimal area element Vector normal to surface	m²
`Q_{\text{enc}}`	enclosed charge Total charge inside Gaussian surface	C

Dimensions : $[L^{3}] [M] [T^{- 4}] {[I]}^{- 1}$

Exemple : A spherical shell with 5 nC total charge has E = 1.44 × 10^4 N/C at 10 cm radius

Magnetostatics: Currents Create Magnetic Fields

Formulas describing how steady currents generate magnetic fields in wires and loops

Biot-Savart Law law

d 𝐁 = \frac{μ_{0}}{4 π} \frac{I d 𝐥 \times \hat{𝐫}}{r^{2}}

Symbole	Signification	Unité
`dB`	infinitesimal magnetic field Contribution from current element	T
`I`	current Steady current in conductor	A
`d\mathbf{l}`	infinitesimal length element Vector along current direction	m
`\hat{\mathbf{r}}`	unit vector from current to point Points from source to field point
`r`	distance from current element Scalar distance	m
`\mu_0`	vacuum permeability 4π × 10^{-7} N/A²

Dimensions : $[M] [T^{- 2}] {[I]}^{- 1}$

Exemple : A 10 A current in a straight wire creates B = 2.0 × 10^{-5} T at 10 cm distance

Ampère's Law law

\oint_{C} 𝐁 \cdot d 𝐥 = μ_{0} I_{enc}

Symbole	Signification	Unité
`B`	magnetic field Integrated along closed path C	T
`d\mathbf{l}`	infinitesimal path element Vector tangent to path	m
`I_{\text{enc}}`	enclosed current Current passing through surface bounded by C	A

Dimensions : $[M] [T^{- 2}] {[I]}^{- 1}$

Exemple : A coaxial cable with 5 A inner current has B = 2.5 × 10^{-6} T at 2 cm radius

Magnetic Field of Long Straight Wire law

B = \frac{μ_{0} I}{2 π r}

Symbole	Signification	Unité
`B`	magnetic field strength Circular field lines around wire	T
`I`	current DC current in wire	A
`r`	radial distance from wire Perpendicular distance	m

Dimensions : $[M] [T^{- 2}] {[I]}^{- 1}$

Exemple : A 15 A current in Wellington's tram wires creates B = 1.5 × 10^{-5} T at 10 cm distance

Electromagnetic Induction: Generating Electricity

Formulas for how changing magnetic fields create electric fields and voltages

Faraday's Law of Induction law

ε = - \frac{d Φ_{B}}{d t}

Formes alternatives

$ε = - N \frac{d Φ_{B}}{d t}$ — For N turns in a coil

Symbole	Signification	Unité
`\varepsilon`	induced EMF Electromotive force driving current	V
`\Phi_B`	magnetic flux $Φ$ _B = $\int$ $𝐁$ $\cdot$ d $𝐀$	Wb
`t`	time Rate of change matters	s

Dimensions : $[M] [L^{2}] [T^{- 3}] {[I]}^{- 1}$

Exemple : A 0.5 m² coil in Christchurch's changing magnetic field (dB/dt = 0.1 T/s) generates $ε$ = 0.05 V

Motional EMF law

ε = B L v

Symbole	Signification	Unité
`\varepsilon`	induced voltage Generated in moving conductor	V
`B`	magnetic field Perpendicular to motion and wire	T
`L`	length of conductor In magnetic field	m
`v`	velocity Perpendicular to both B and L	m/s

Dimensions : $[M] [L^{2}] [T^{- 3}] {[I]}^{- 1}$

Exemple : A 30 cm metal rod moving at 2 m/s through 0.8 T field in Dunedin generates $ε$ = 0.48 V

Lenz's Law law

ε_{induced} = - \frac{d Φ_{B}}{d t}

Symbole	Signification	Unité
`\varepsilon_{\text{induced}}`	induced EMF polarity Negative sign indicates direction opposes change	V
`\Phi_B`	magnetic flux Same as Faraday's Law	Wb

Dimensions : $[M] [L^{2}] [T^{- 3}] {[I]}^{- 1}$

Exemple : When magnet falls into coil in Hamilton, induced current creates field opposing magnet's motion

Lorentz Force: Charges in Fields

The force experienced by moving charges in electric and magnetic fields

Lorentz Force Law law

𝐅 = q (𝐄 + 𝐯 \times 𝐁)

Formes alternatives

$F_{x} = q E_{x}$ — Electric force component
$F_{y} = q v_{x} B_{z}$ — Magnetic force component when v ⊥ B

Symbole	Signification	Unité
`F`	Lorentz force Total electromagnetic force on charge	N
`q`	electric charge Can be positive or negative	C
`E`	electric field At charge's position	N/C
`v`	charge velocity Vector	m/s
`B`	magnetic field At charge's position	T

Dimensions : $[M] [L] [T^{- 2}]$

Exemple : An electron (q = -1.6×10^{-19} C) moving at 10^6 m/s perpendicular to 0.1 T field in a Dunedin lab experiences F = 1.6×10^{-14} N

Force on Current-Carrying Wire law

F = I 𝐋 \times 𝐁

Symbole	Signification	Unité
`F`	magnetic force On wire segment	N
`I`	current Steady current	A
`L`	wire length vector Direction same as current	m
`B`	magnetic field External field	T

Dimensions : $[M] [L] [T^{- 2}]$

Exemple : A 20 cm wire carrying 5 A in a 0.3 T field in Christchurch experiences F = 0.3 N

Cyclotron Frequency definition

ω = \frac{| q | B}{m}

Symbole	Signification	Unité
`\omega`	angular frequency Circular motion frequency	rad/s
`q`	charge magnitude Absolute value	C
`B`	magnetic field Perpendicular to velocity	T
`m`	particle mass e.g. electron mass 9.11×10^{-31} kg	kg

Dimensions : $[T^{- 1}]$

Exemple : A proton (m = 1.67×10^{-27} kg) in 1 T field at University of Otago cyclotron has f = 15.2 MHz

Maxwell's Equations: The Big Picture

The four fundamental equations governing all classical electromagnetism

Gauss's Law for Electricity law

\oint_{S} 𝐄 \cdot d 𝐀 = \frac{Q_{enc}}{ε_{0}}

Symbole	Signification	Unité
`E`	electric field Integrated over closed surface	N/C
`Q_{\text{enc}}`	enclosed charge Total charge inside	C

Dimensions : $[L^{3}] [M] [T^{- 4}] {[I]}^{- 1}$

Exemple : A 10 nC point charge at Auckland University creates flux 113 N·m²/C through 1 m radius sphere

Gauss's Law for Magnetism law

\oint_{S} 𝐁 \cdot d 𝐀 = 0

Symbole	Signification	Unité
`B`	magnetic field Integrated over closed surface	T
`dA`	area element No magnetic monopoles exist	m²

Dimensions : $[M] [T^{- 2}] {[I]}^{- 1}$

Exemple : Magnetic flux through any closed surface in Christchurch is always zero

Faraday's Law law

\oint_{C} 𝐄 \cdot d 𝐥 = - \frac{d Φ_{B}}{d t}

Symbole	Signification	Unité
`E`	induced electric field Non-conservative field	N/C
`\Phi_B`	magnetic flux Through surface bounded by C	Wb

Dimensions : $[M] [L^{2}] [T^{- 3}] {[I]}^{- 1}$

Exemple : Changing B field in Wellington's power grid induces E field that drives current

Ampère-Maxwell Law law

\oint_{C} 𝐁 \cdot d 𝐥 = μ_{0} I_{enc} + μ_{0} ε_{0} \frac{d Φ_{E}}{d t}

Symbole	Signification	Unité
`B`	magnetic field Integrated along path C	T
`I_{\text{enc}}`	enclosed current Conduction current	A
`\Phi_E`	electric flux Through surface bounded by C	V·m

Dimensions : $[M] [T^{- 2}] {[I]}^{- 1}$

Exemple : Displacement current in charging capacitor at Canterbury University creates magnetic field

Electrostatics: The Invisible Push and Pull

Magnetostatics: Currents Create Magnetic Fields

Electromagnetic Induction: Generating Electricity

Lorentz Force: Charges in Fields

Maxwell's Equations: The Big Picture

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