electricityandmagnetism.md

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Electricity and Magnetism

Electrical conductivity

$σ = 1/ρ$ ,

where

• $ρ$ is the electrical resistivity, $[Ω·m]$.

Electrical resistivity

$ρ = R·(A/L)$ ,

where

• $R$ is the electrical resistance, $[Ω]$,
• $L$ is the length of the specimen, $[m]$,
• $A$ is the cross-sectional area of the specimen, $[m^2]$.

Electrostatic force and electric fields

$E = \displaystyle\frac {F}{q}$ ,

where

• $E$ is the electric field strength, $[N/C]$,
• $F$ is the force on the charged particle, $[N]$,
• $q$ is the charge on the object experiencing the force, $[C]$.

Vacuum permittivity

$ε_0 = 1/(4⋅π⋅k_e)$ ,

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$.

Electric field of a point charge

$E = k_e⋅\displaystyle \left(\frac{|Q|}{r^2}\right)$ ,

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$,
• $Q$ is the value of point charge, $[C]$,
• $r$ is the distance from the point charge, $[m]$.

Electric field of conducting sphere

The electric field of a conducting sphere with charge $Q$ is,

$E = k_e⋅\displaystyle \left(\frac{|Q|}{r^2}\right)$ , if $r>R$ ,

$E = 0$ , if $r < R$

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$,
• $Q$ is the value of charge, $[C]$,
• $r$ is the distance from the sphere, $[m]$.
• $R$ is the sphere radius, $[m]$.

Electric field of uniform charge sphere:

$E = k_e⋅\displaystyle\left(\frac{|Q|}{r^2}\right)$

if $\displaystyle r > R$ , and

$E = k_e⋅\displaystyle\left[|Q|⋅\left(\frac{r}{R^3}\right)\right]$

if $\displaystyle r < R$

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$,
• $Q$ is the value of charge, $[C]$,
• $r$ is the distance from the sphere, $[m]$.
• $R$ is the sphere radius, $[m]$.

Electric Field of a single conducting plate

$E = \displaystyle\frac{\sigma}{2⋅\epsilon_0⋅\epsilon_r} = \left(\frac{1}{2⋅\epsilon_0⋅\epsilon_r}\right)⋅\left(\frac{|Q|}{S}\right)$ ,

where

• $\epsilon_0$ is the vacuum permittivity, $[C^2/N/m^2]$,
• $\epsilon_r$ is the relative permittivity, a dimensionless number,
• $Q$ is the value of plate charge, $[C]$,
• $S$ is the area of plate charge, $[m^2]$.

Electric field of parallel conducting plates

$E = \displaystyle\frac{\sigma}{\epsilon_0⋅\epsilon_r} = \left(\frac{1}{\epsilon_0⋅\epsilon_r}\right)⋅\left(\frac{|Q|}{S}\right)$ ,

where

• $\epsilon_0$ is the vacuum permittivity, $[C^2/N/m^2]$,
• $\epsilon_r$ is the relative permittivity, a dimensionless number,
• $Q$ is the charge of single plate, $[C]$,
• $S$ is the area of single plate, $[m^2]$.

Electric field of line charge

$E = \displaystyle\left(\frac{1}{2⋅\epsilon_0⋅\epsilon_r}\right)⋅\left(\frac{\lambda}{R}\right)=\left(\frac{1}{2⋅\epsilon_0⋅\epsilon_r}\right)⋅\left(\frac{|Q|}{L⋅R}\right)$ ,

where

• $\epsilon_0$ is the vacuum permittivity, $[C^2/N/m^2]$,
• $\epsilon_r$ is the relative permittivity, a dimensionless number,
• $Q$ is the line charge, $[C]$,
• $L$ is the line length, $[m]$,
• $r$ is the distance from the wire, $[m]$.

Electrostatic force between two point charges

$F_e = k_e·(q_1·q_2)/r^2$ ,

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$,
• $q_1$ is the point charge 1, $[C]$,
• $q_2$ is the point charge 2, $[C]$,
• $r$ is the distance between the charges, $[m]$.

Electrostatic potential energy

$U_e = k_e·(q_1·q_2)/r$ ,

where

• $k_e$ is the Coulomb constant, $k_e ≈ 8.988×10^9$ $[N⋅m^2/C^2]$,
• $q_1$ is the charge 1, $[C]$,
• $q_2$ is the charge 2, $[C]$,
• $r$ is the distance between the charges.

Lorentz force: electric charge

$F = q⋅[v⋅B⋅sin(α)]$ ,

where

• $F$ is the module of Lorentz force, $[N]$,
• $q$ is the charge of particle $q$, $[C]$,
• $v$ is the velocity of the charge, $[m/s]$,
• $B$ is the magnetic field, $[T]$,
• $α$ is the angle between $v$ and $B$, $[rad]$.

Lorentz force: electric current

$F = i⋅[l⋅B⋅sin(α)]$ ,

where

• $F$ is the module of magnetic force, $[N]$,
• $i$ is the value of electric current, $[A]$,
• $l$ is the length of the line, $[m]$,
• $B$ is the magnetic field, $[T]$,
• $α$ is the angle between $l$ and $B$, $[rad]$.

Biot-Savart law: magnetic field created by a long straight current-carrying wire

$B = \displaystyle\frac{μ_0}{2π}⋅\left(\frac{I}{r}\right)$ ,

where

• $B$ is the magnetic field, $[T]$,
• $μ_0$ is the vacuum magnetic permeability, $μ_0=4π⋅10^{-7}$ $[T⋅m/A]$,
• $I$ is the current intensity flowing in the long wire, $[A]$,
• $r$ is the distance of the magnetic field from the wire, $[m]$.

Biot-Savart law: magnetic field produced by a current-carrying circular loop

$B = \displaystyle\frac{μ_0}{2}⋅\left[I/\left[\frac{(z^2+R^2)^{3/2}}{R^2}\right]\right]$ ,

where

• $B$ is the magnetic field, $[T]$,
• $μ_0$ is the vacuum magnetic permeability, $μ_0=4π⋅10^{-7}$ $[T⋅m/A]$,
• $I$ is the current intensity flowing in the circular loop, $[A]$,
• $R$ is the radius of circular loop, $[m]$,
• $z$ is the distance along circular loop axis from center, $[m]$.

Biot-Savart law: magnetic field produced by a current-carrying solenoid

$B = \displaystyle\ {μ_0}⋅{N}⋅\left(\frac{I}{L}\right)$ ,

where

• $B$ is the magnetic field strength inside a solenoid, $[T]$,
• $μ_0$ is the vacuum magnetic permeability, $μ_0=4π⋅10^{-7}$ $[T⋅m/A]$,
• $N$ is the number of loops, a dimensionless number,
• $I$ is the current intensity flowing in the solenoid, $[A]$,
• $L$ is the length of the solenoid, $[m]$.

Forces between parallel conductors

$F = \displaystyle\left[ \frac{μ_0}{2\pi}⋅\left(\frac{L}{r}\right)\right]⋅(i_1⋅i_2)$ ,

where

• $F$ is the force between the conductors, $[N]$,
• $μ_0$ is the vacuum magnetic permeability, $μ_0=4π⋅10^{-7}$ $[T⋅m/A]$,
• $i_1$ is the current in wire 1, $[A]$,
• $i_2$ is the current in wire 2, $[A]$,
• $r$ is the distance separating the conductors, $[m]$,
• $L$ is the length of the conductors, $[m]$.

Magnetic flux

$\Phi_b = B⋅S⋅cos(\alpha)$ ,

• $B$ is the magnitude of the magnetic field, $[T]$,
• $S$ is the area of the surface, $[m^2]$,
• $\alpha$ is the angle between the magnetic field lines and the normal to $S$, $[rad]$.

Faraday's law of induction and Lenz's law

$\epsilon = -\displaystyle\left(\frac{\Delta\Phi_B}{\Delta t}\right)⋅N$ ,

where

• $\epsilon$ is the induced electromotive force, $[V]$,
• $\Delta\Phi_B$ is the in change magnetic flux, $[W_b]$,
• $\Delta t$ is the change in time, $[s]$,
• $N$ is the number of turns in a coil, a dimensionless number.

Line integral of the magnetic field

$C_{L}\left(B\right) = \displaystyle\sum_{i=1}^{n} B_i⋅l_i⋅cos(\alpha)$ ,

where

• $C_L(B)$ is line integral of magnetic field, $[T⋅m]$,
• $B_i$ is the value of magnetic field on the element length $l_i$, $[T]$,
• $l_i$ is the length of element $i$,
• $\alpha$ is the angle betweeb $B_i$ and element length $l_i$, $[rad]$.

Displacement current

$i_d = \displaystyle\frac{\epsilon_0⋅\Delta\Phi_E}{\Delta t}$

where

• $i_d$ is the Maxwell's Displacement current, $[A]$;
• $\epsilon_0$ is the vacuum permittivity, $[C^2/N/m^2]$;
• $\Delta\Phi_E$ is the change in electric flux, $[V⋅m]$;
• $\Delta t$ is the change in time, $[s]$.

Ampère's circuital law (with Maxwell's addition)

$C_L(B) = \displaystyle\mu_0⋅i + \mu_0⋅\left[\frac{\epsilon_0⋅\Delta\Phi_E}{\Delta t}\right]$

where

• $μ_0$ is the vacuum magnetic permeability, $μ_0=4π⋅10^{-7}$ $[T⋅m/A]$;
• $i$ is the electric current, $[A]$;
• $\epsilon_0$ is the vacuum permittivity, $[C^2/N/m^2]$;
• $\Delta\Phi_E$ is the change in electric flux, $[V⋅m]$;
• $\Delta t$ is the change in time, $[s]$.

Inductance

$L = \displaystyle\frac{\Delta\Phi_B}{\Delta i}$ ,

where

• $L$ is the inductance, $[H]$,
• $\Delta\Phi_B$ is the change in magnetic flux, $[T⋅m]$,
• $\Delta i$ is the change in electric curremt, $[A]$.

RL circuit

$i(t) = \displaystyle\frac{V}{R}⋅(1-e^{-t⋅ R/L})$ ,

where

• $i(t)$ is the value of current, $[s]$,
• $V$ is the electric potential, $[V]$,
• $R$ is the value of resistance, $[\Omega]$,
• $L$ is the circuit inductance, $[H]$.

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