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Electromagnetism

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Magnetism

Magnetic field

Vector field demonstrating the magnetic forces applied in a space

Magnetic flux

Scalar quantity \(\Phi\) representing the total magnetic field through the surface area of a material

\(\Phi = \iint_{S}\textbf{B} \cdot d\textbf{S} \)

• \(\textbf{B}\) is the magnetic field
• \(S\) is a differentiable surface

Biot-Savart law

Law representing the magnetic field of a current travelling up the z-axis in cylindrical coordinates. This representation of a magnetic field is the base from which Maxwell's equations are derived.

\( \textbf{B} = \frac{\mu_0 Q}{4 \pi \rho} \hat{\theta} \)

Electricity

Electric charge Carica elettrica 電荷 (q)

Scalar quantity \(q\) representing the physical property in material's matter that causes a force when exposed to an electric field. An electron holds a charge \(-1e\), while a proton holds a charge \(+1e\); electric charge is therefore quantized. It is also conserved

Elementary charge (\(e\))

Physical constant that represent the quantity of charge held by a proton

\(e=1.6 \times 10^{-19}C\)

Charge density

Charges per unit of volume

\(\rho = \frac{Q_{\text{enc}}}{V}\)

Electrode

Cathode; electrode which electrons flow into

Anode; electrode which electrons flow towards

Electric field

Vector field demonstrating the forces applied to charged particles

Current Corrente 電流 (I)

Scalar quantity \(I\) representing the toral rate of charge flow through a surface

\( I = \frac{dq}{dt} \)

Electron flow

Current is the flow of electrons from a place of lower charge (anode) to a place of higher charge (cathode)

Conventional current

Arbitrarily defining the flow of current from cathode to anode for notational purposes

Current density

Vector representing the direction and magnitude of charge flow per a unit of area

\(\textbf{J}= \frac{I_{\text{enc}}}{A}\)

Electric potential Potenziale elettrico 電位 (V)

Scalar quantity \( V \) representing the work required to move one single charge from a reference point to a certain point in an electric field. This manifests the phenomenon of points with higher charges requiring less energy to move to.

\(\displaystyle V (\textbf{r}_1) = - \int_{\mathcal{C}} \textbf{E} \cdot d\textbf{r} \)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(\textbf{r}_0\) is a vector to the starting reference point
• \(\textbf{r}_1\) is a vector to the desired destination point
• \(\mathcal{C} \subset X\) is a differentiable curve in the space representing the path from \(\textbf{r}_0\) to \(\textbf{r}_1\)
• \(\textbf{r} : [0,t] \to \mathcal{C}\) is the position function of the electric charge along \(\mathcal{C}\) (note that \(\textbf{r}_1 = \textbf{r}(0)\) and \(\textbf{r}_1 = \textbf{r}(t) \) )
• \(d\textbf{r} = \textbf{r}'(t)dt\) is the infinitesimal change in the electric charge's position

Electric potential energy

Scalar quantity \( U \) representing the work required to move a charge of magnitude \(q\) from a reference point to a certain point in an electric field.

\(U(\textbf{r}_1) = q V(\textbf{r}_1) \)

Conservation of energy

Conservation of charge

Voltage Voltaggio 電圧(V)

Scalar quantity \( \Delta V \) representing the difference of electric potential energy of two points; comparing two points with eachother rather than comparing them both with the reference point

Since like charges repel and dislike charges attract, the difference in charge causes current to occur.

\( \Delta V(\textbf{r}_1 , \textbf{r}_2) = V( \textbf{r}_2 ) - V( \textbf{r}_1 ) \)

Notationally, \( \Delta V \) can be shortened to \(V\) (since \( \textbf{r}_1 \text{ is the chosen reference point } \implies \Delta V(\textbf{r}_1 , \textbf{r}_2) = V(\textbf{r}_2 )\) )

Resistance Resistanza 抵抗 (R)

Scalar quantity \(R\) representing a material's resistance to current

\(R = \frac{\rho \ell}{A}\)

• \(\rho\) is the resistivity of the medium
• \(\ell\) is the length of the medium
• \(A\) is the cross sectional area of the medium

Conductance コンダクタンス (G)

Scalar quantity \(G\) representing a material's propensity to conduct changes throughout itself

\(G = \frac{1}{R}\)

Conductor

Insulator

Ohm's law

Empirical law stating proportionality between current and voltage for ohmic materials. This implies the corrolary that resistance is constant in ohmic materials

\(V = IR\)

Energy (electrical) Energia (elettrica) エネルギー（電気的） (E)

Scalar quantity \(E\) representing energy created by current that can be transferred into work

\( E=qV(\textbf{r}_1) \)

• \(q\) is the charge magnitude
• \(\textbf{r}_1\) is the desired destination position for the charge to arrive at

Power (electrical) Potenza (elettrica) 電力（電気的） (P)

Scalar quantity \(P\) representing rate at which energy is transferred

\( P = \frac{dE}{dt}\)

\( P= \frac{dE}{dt}= \frac{dq}{dt}V=IV \)

\( P=IV=I^{2}R=\frac{V^2}{R} \)

\( E= \int_{t_1}^{t_2} P(t) dt \)

Capacitance Capacitanza 容量 (C)

Scalar quantity \(C\) representing material's ability to store electric charges per volt

\( C = \frac{q}{V}\)

• \(q\) is charge
• \(V\) is voltage
• \(C\) is capacitance

Permittivity

Scalar quantity \(\varepsilon\) that describes how easily a material polarises with an applied voltage, how easily polarizing is permitted to occur

Permittivity of free space

Permittivity of a vacuumm denoted \(\varepsilon_0\)

\(\varepsilon_{0} = 8.8541878128(13) \times 10^{-12} F/m\)

Permeability

Scalar quantity \(\mu \) that describes the amount of magnetisation in an object when a magnetic field is applied to it, how much a magnetic field permeates in a material

Permeability of free space

Permeability of a vacuumm denoted \(\mu_0\)

\(\mu_{0} = 4\pi \times 10^{-7} H/m\)

Induction

Phenomenon of change in a magnetic field across an electrical conductor producing a voltage called Electromotive Force (EMF). This is the basis of Faraday's law, and relates the fields of magnetism and electricity.

Inductance Induttanza インダクタンス (L)

Scalar quantity \(L\) representing a material's ability to form a magnetic field per ampere

\( L = \frac{\Phi}{I} \)

• \(\Phi\) is the magnetic flux of a closed surface
• \(I\) is the current through the inductor
• \(L\) is the inductance

Coulomb's law

Inverse square law representing the electric field of one particle centered at the origin in spherical coordinates. This representation of an electric field is the base from which Maxwell's equations are derived.

\( \textbf{E} = \frac{Q}{4 \pi \varepsilon_{0} \rho^2} \hat{\rho} \)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(\varepsilon_0\) is the vacuum permittivity
• \(Q\) is the charge of the electric charge exerting the force

An electric charge of density \(q\) is therefore subject to the following force:

\( \textbf{F}= q\textbf{E} \)

Electric fields representing the superposition of multiple electric charges can be accounted for by summing the electric fields of each electric charge

Maxwell equations

Set of differential equations governing classical electromagnetism by stating the properties of electic and magnetic fields that are either derived from Coulomb's law and Biot-Savart law and an empirical observation (Faraday's law).

Differential

• \( \nabla \cdot \textbf{E} = \frac{\rho(\textbf{r}(t)}{\varepsilon_0} \) (Gauss' law)
• \( \nabla \cdot \textbf{B} = 0 \) (Gauss' magnetism law)
• \(\nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} \) (Faraday's law)
• \( \nabla \times \textbf{B} = \mu_0 \textbf{J}(\textbf{r}(t)) + \mu_0 \varepsilon_0 \frac{\partial \textbf{E}}{\partial t}\) (Ampere's law)

Integral

Gauss' law

Law stating that the divergence of the electric flux density is the electric charge density

Differential

\( \nabla \cdot \textbf{E} = \frac{\rho(\textbf{r}(t)}{\varepsilon_0} \)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(\rho\) is the electric charge's densitiy at position \(\textbf{r}(t)\)
• \(\varepsilon_0\) is the vacuum permittivity
• \(d\textbf{r} = \textbf{r}'(t)dt\) is the infinitesimal change in the electric charge's position
• \(\mathcal{C} \subset X\) is a differentiable curve in the space representing the path of the electric charge
• \(\textbf{r}(t) : [t_0,t_1] \to \mathcal{C}\) is the position function of the electric charge along \(\mathcal{C}\)

Integral

\( \displaystyle \iint_{S} \textbf{E} \cdot d\textbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0} \)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(Q_{\text{enc}}\) is
• \(\varepsilon_0\) is the vacuum permittivity
• \(d\textbf{r} = \textbf{r}'(t)dt\) is the infinitesimal change in the electric charge's position
• \(\mathcal{C} \subset X\) is a differentiable curve in the space representing the path of the electric charge
• \(\textbf{r}(t) : [t_0,t_1] \to \mathcal{C}\) is the position function of the electric charge along \(\mathcal{C}\)

Gauss' magnetism law

Law stating magnetic fields have zero divergence, they are conserved under flux

Differential

\( \nabla \cdot B = 0 \)

Integral

\( \displaystyle \iint_{S} \textbf{B} \cdot d\textbf{S} = 0 \)

Faraday' law

Law stating that the change in a magnetic field produces a change in an electric field's curl (work to move an electric charge around electric field) of opposite polarity

Law stating that electromotive force (curl of electric field) equals the magnetic field induced by

Note that using the Coulomb's law and taking the curl returns \(0\), however this experimental law makes the mathematical structure more realistic

Differential

\(\nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} \)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(\textbf{B} : \mathbb{R}^n \to \mathbb{R}^n\) is the magnetic vector field function

Integral

\(\displaystyle \oint_{\mathcal{\partial S}} \textbf{E} \cdot d\textbf{r} = - \iint_{S} \frac{\partial \textbf{B}}{\partial t} \cdot d\textbf{S}\)

• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric vector field function
• \(\textbf{B} : \mathbb{R}^n \to \mathbb{R}^n\) is the magnetic vector field function
• \(\partial S\) is the boundary of the differentiable surface

Ampere' law

Law stating that the curl of magnetic field

Differential

\( \nabla \times \textbf{B} = \mu_0 \textbf{J}(\textbf{r}(t)) + \mu_0 \varepsilon_0 \frac{\partial \textbf{E}}{\partial t}\)

• \(\textbf{B} : \mathbb{R}^n \to \mathbb{R}^n\) is the magnetic field
• \(\textbf{E} : \mathbb{R}^n \to \mathbb{R}^n\) is the electric field
• \(\textbf{J}\) is the current density inducing the magnetic field
• \(\mu_0\) is the vacuum permeability

Integral

\( \displaystyle \oint_{\mathcal{C}} \textbf{B} \cdot d\textbf{r} = \mu_0 I_{\text{enc}} \)

• \(\textbf{B} : \mathbb{R}^n \to \mathbb{R}^n\) is the magnetic vector field function
• \(I_{\text{enc}}\) is the enclosed current
• \(\mu_0\) is the vacuum permeability
• \(d\textbf{r} = \textbf{r}'(t)dt\) is the infinitesimal change in the electric charge's position
• \(\mathcal{C} \subset X\) is a differentiable curve in the space representing the path of the electric current
• \(\textbf{r}(t) : [t_0,t_1] \to \mathcal{C}\) is the position function of the electric charge along \(\mathcal{C}\)

Historical note

Lorentz force law

Law asserting the force of a charged particle with velocity \(\textbf{v}\) amidst an electric and magnetic field

\(\textbf{F} = q(\textbf{E} + \textbf{v} \times \textbf{B})\)

• \(q\) is the charge of the particle
• \(\textbf{v}\) is the velocity vector of the particle
• \(\textbf{E}\) is the electric field
• \(\textbf{B}\) is the magnetic field
• \(\textbf{F}\) is the force upon the charged particle

Skin effect

Tendency of current density to be maximised on a surface and exponentially decaying approaching the center