𝜑𝖤. Electricity - JulTob/Mathematics GitHub Wiki

Electricity

Electricity is really just organized lightning.
— George Carlin

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🌌 Electromagnetism: Unveiling the Forces of Nature 🌩️

By María Jesús Cuesta

"Electricity is really just organized lightning." — George Carlin

Vector Calculus Refresher: The Building Blocks 🔧

Vector Components

Every vector can be broken down into its components along the x, y, and z axes. Let's call them:

$A_x, A_y, A_z$

Magnitude of the Vector

The magnitude is like finding out how long your vector is when stretched across space.

|A| = \sqrt{A_x^2 + A_y^2 + A_z^2}

Unit Vectors

For simplicity, we use unit vectors along each axis:

$u_x = \hat{i}$ (for the x-axis)
$u_y = \hat{j}$ (for the y-axis)
$u_z = \hat{k}$ (for the z-axis)

Transformation of Coordinates 🧭

To shift from one coordinate system to another, we use transformation matrices. It’s like changing your point of view.

Matrix Transformation

Here’s the magic matrix that helps us switch coordinates for vector $A$:

A = A_x \cdot u_x + A_y \cdot u_y + A_z \cdot u_z

And when you need to rotate or transform, you use the transformation matrix:

\begin{pmatrix}
cosu & senu & 0 \\
-senu & cosu & 0 \\
0 & 0 & 1
\end{pmatrix}

🎯 This shifts your vector by applying the transformation rules to each component.

Spherical Coordinate System 🌐

In some cases, we work with vectors using spherical coordinates instead of Cartesian.

Magnitude in Spherical Coordinates

Here, the magnitude $|A|$ is simpler, represented by just $r$.

|A| = r

Where $r$ is the radius or the distance from the origin in this new view.

Electric Charge ⚡: The Invisible Glue of the Universe

Charge is like an invisible force field that particles carry around, attracting or repelling other charges based on whether they’re similar or different.

Electrons carry a negative charge, with $e^- = 1.6 \cdot 10^{-19}$ Coulombs.
Protons carry a positive charge, but the same magnitude as the electron.

Force Between Charges: Coulomb's Law

The force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them:

F = k_c \cdot \frac{q_1 \cdot q_2}{r^2}

Where:

$k_c$ is Coulomb's constant.
$q_1, q_2$ are the magnitudes of the charges.
$r$ is the distance between them.

Quantization of Charge

Charge is quantized, meaning it only comes in multiples of the elementary charge ($e$).

Electric Fields: The Charge's Reach 🌍

The electric field ($E$) is like an aura around a charge, representing the force it exerts on other charges:

E = \frac{F}{q}

Where $E$ is the force per unit charge, measured in Newtons per Coulomb (N/C).

Dielectric Constant (aka $\epsilon_0$) 🧲

Not all materials respond the same way to electric charges. To quantify this, we introduce epsilon ($\epsilon$), the dielectric constant, which represents a material’s ability to conduct electric fields.

In a vacuum (the simplest case):

\epsilon_0 = 8.85 \cdot 10^{-12} \ \text{F/m}

Relationship Between Coulomb’s Constant and Epsilon

To connect them:

k = \frac{1}{4 \cdot \pi \cdot \epsilon_0}

Continuous Charge Distributions 🌫️

When charges are spread out rather than isolated, we talk about continuous distributions. For this, the electric field is calculated using integrals over the distribution.

Steps for finding the electric field $E$:

Calculate individual contributions.
Calculate the total magnitude.
Determine the direction ($\hat{r}$).
Solve the integral over the charge distribution.

Coulomb's Law: Recap

Coulomb's law isn't just for point charges. For distributions, we integrate it over all tiny bits of charge.

Protons have a positive fundamental quality called $\color{#E77728}charge$. Electrons have an equal but negative charge.

\color{#E77728}
Q = +P·Ⓟ_𝚚 -N·ⓔ_𝚚

\color{#EDB230}
Ⓟ_𝚚 = -ⓔ_𝚚 = 1.602·⏨(-19) ⠀[C]

Coulomb's Law

Charges exercise forces on each other.

\color{#7DCE82}
𝐹 = |\overrightarrow{𝐹}| = 𝑘 \frac{
{\color{#EDB230}|𝑄|}
{\color{#E77728}|𝚚|}
}{\color{#00FFF5}𝚛²}

⚡

$\color{#7DCE82}𝑘$: Electric constant. Balances⠀units. Material⠀dependent.

\color{#7DCE82}
\begin{matrix}
𝑘 =  &   8.99 Ⅹ⁹ [N · m² : C²] & In⠀air / vacuum \\
 &   1 : 4 𝜋 𝜀 &  
\end{matrix}

\color{#00FFF5}
\begin{matrix}
𝜀₀ =  & 8.85 Ⅹ⁻¹²  & [C : N m²]	&  In⠀air / vacuum
\end{matrix}

Electric Field

Electric force on a test charge 𝚚

\color{#1BCA07}
𝔼⃗ = \frac{𝐹⃗}{\color{#E77728}𝚚} ⠀⠀   [N:C]

The point field for a charge $\color{#EDB230}𝑄$

\color{#1BCA07}
𝔼⃗☉ = 𝑟̂ 𝑘 \frac{{\color{#EDB230}𝑄}}{\color{#00FFF5} 𝑟² }

The field near of a charged plate Area 𝐴 with a charge 𝑄

𝛔 = 𝑄 : 𝐴
𝔼⃗🝞 = 𝐴̂ 𝛔 : 2𝜀

And In a capacitor plaques

𝔼␥ = 𝛔 : 𝜀

Capacitancy ⫩

𝑄 = Ⅴ·Ⅽ

Current

𝐈 = ∂𝑞 : ∂𝑡

Resistance ┤╲╱╲╱╲╱╲├

𝐑 = 𝐈∶𝐕