Electromagnetism

Electromagnetism is a fundamental branch of physics that studies the interaction between electric charges and magnetic fields. It is one of the four fundamental forces of nature, alongside gravity, the weak nuclear force, and the strong nuclear force. Electromagnetism encompasses both electricity and magnetism, which were once thought to be separate phenomena but are now understood to be deeply interconnected.

Below you can find the main definitions, necessary to understand the basics of electromagnetism.

The electric field is a vector field generated by any charged particle in space or matter. It is associated with an electrostatic force that every charged particle exerts on any other charged particle, following the so-called Coulomb's law: $$F = \frac{k_e}{\varepsilon} \frac{q_1 q_2}{d^2} = \frac{1}{4\pi \varepsilon \varepsilon_0} \frac{q_1 q_2}{d^2}$$ An electric field exerts both attractive and repulsive forces, according to the signs of the two interacting charges.

An electric field can also be generated through a varying magnetic field passing through a defined area; this phenomenon is described by Faraday's induction law: $$\oint_{L} \mathbf E \cdot d \mathbf l = -\frac{d}{dt} \int_{S} \mathbf B \cdot d \mathbf S = -\frac{d}{dt} \Phi(\mathbf S)$$

The magnetizing field (also known as magnetic field) is a vector field generated by a moving electrically charged particle in space or matter. Any oscillating, moving, flowing particle of a group of particles generates a static or dynamic magnetic field; following Faraday's induction law, if the magnetizing field is dynamic, it could induce an electric field in surrounding matter.

The magnetizing field is an absolute quantity generated by moving charged particles; however, when one wants to study the interaction between magnetic field and matter, it is good to use the concept of magnetic induction.

The electric induction (or electric displacement field) is a vector field that sums up the effect of electric polarization in a material and the external electric field. It is used to study the interaction between electric fields and matter, especially those materials that contain polarized charges inside. It is defined as: $$\mathbf D = \varepsilon_0 \mathbf E + \mathbf P$$ where P is the polarization density (the amount of electric dipole moment over a defined volume) and E is the external electric field. P is defined as: $$\mathbf P = \frac{\Delta \mathbf p}{\Delta V}$$ where V is the volume in [m³] and p is the dipole moment, in [C m], defined as: $$\mathbf p = q \cdot d$$ where q is the modulus of the two electric charges creating the electric dipole (equal in modulus and opposite in sign) in [C], and d is the distance between the two in [m].

As you can figure out, there also exists a magnetic induction field: it expresses how a material responds to an external magnetizing field, and it is directly related to the intrinsic properties of the material. Similarly to electric induction, magnetic induction is defined as: $$\frac{1}{\mu_0} \mathbf B = \mathbf H + \mathbf M$$ where M is the magnetization vector (the magnetic moment over a defined volume) and H is the external magnetizing field. M is defined as: $$\mathbf M = \frac{\Delta \mathbf m}{\Delta V}$$ where V is the volume in [m³] and m is the magnetic moment, in [A m²], defined as: $$\mathbf m = p \cdot \mathbf l \ or \ \mathbf m = I \cdot \mathbf S$$ where p is the magnetic pole strength, l is the distance between the two poles for the first equation and I is the current running along a closed loop and S is the surface enclosed by that loop for the second equation. In the second case, the vector representing the moment (that has the same direction as the surface vector) is the one normal to the surface itself.

Symbols are defined as follows $$E \ electric \ field \ \left[ \frac{V}{m} \right] \qquad H \ magnetizing \ field \ \left[ \frac{A}{m} \right] $$ $$D \ electric \ induction \ \left[ \frac{C}{m^2} \right] \qquad B \ magnetic \ induction \ \left[ T \right],\left[ \frac{Wb}{m^2} \right]$$ $$P \ polarization \ density \ \left[ \frac{C m}{m^3} \right] \qquad M \ magnetic \ moment \ \left[ \frac{A m^2}{m^3} \right]$$ $$\rho \ charge \ density \ \left[ \frac{C}{m^3} \right] $$ $$J \ total \ current \ density \ (induced \ - \ eddy\ currents \ included) \ \left[ \frac{A}{m^2} \right]$$

Assuming rho=0 and J=0, Maxwell's equations leads to $$\nabla \cdot \mathbf E = 0, \nabla \times \mathbf E = - \frac{\partial \mathbf B}{\partial t}$$ $$\nabla \cdot \mathbf B = 0, \nabla \times \mathbf B = \mu_0 \varepsilon_0 \frac{\partial \mathbf E}{\partial t}$$ which lead to $$\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf E}{\partial t^2} = \nabla^2 \mathbf E$$ $$\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf B}{\partial t^2} = \nabla^2 \mathbf B$$ The phase velocity of light is, generically $$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0 \mu_r \varepsilon_r}}$$

This vector represents, in a three-dimensional space, the direction and the modulus of the energy flux of an electromagnetic wave. It is expressed in [W m²] and defined as: $$\mathbf S = \mathbf E \times \mathbf H$$ Instantaneous power is defined as $$\mathbf S(t) = \mathbf E(t) \times \mathbf H(t) = $$

F is the force a 0-dimension electric particle is subjected to, q is the value of the elementary charge, E is the external electric field, v is the speed of the particle and B is the external magnetic induction $$\mathbf F = q (\mathbf E + \mathbf v \times \mathbf B)$$

F is the force a 1-dimension wire is subjected to, I is current intensity, l is the length of the wire and B is the external magnetic induction $$\mathbf F = I \mathbf l \times \mathbf B$$

F is the force between two 1-dimension wires, I₁ and I₂ are the two current intensities, l is the length of the two wires and dis is the distance between the two $$\mathbf F = \frac{\mu_0}{2 \pi} I_1 I_2 \frac{l}{d}$$

The wire ideally has 1 dimension only; the current is AC or DC; in the case of AC, all parasitic phenomena are neglected (eddy currents, skin effect). I is the current intensity, d is the distance between the wire and the measurement point $$B = \frac{\mu_0 I}{2 \pi d}$$

Parasitic phenomena are neglected; R is the radius of the spire, I is the current intensity. The material inside the spire could be a vacuum or a linear material; no saturation nor hysteresis effect is considered by this formula. $$B = \frac{\mu_0 \mu_r I}{2R}$$

Parasitic phenomena are neglected; N is the number of turns, l is the total length of the solenoid, I is the current intensity. The material inside the solenoid could be a vacuum or a linear material; no saturation nor hysteresis effect is considered by this formula. $$B = \frac{\mu_0 \mu_r N I}{l}$$