Summary of Electrostatics and Electromagnetism
Electrostatics and Electromagnetism Explained for Students
Introduction
Electrostatics studies electric charges at rest, the forces between them, and the electric fields and potentials they produce. This material condenses core concepts from electric charge and Coulomb's law through Gauss's law and basic capacitance, with worked examples and practical applications suitable for university-level study.
Chapter 1: Electric Charge and Coulomb's Law
Basic properties
Definition: Electric charge is an intrinsic property of matter that causes it to experience a force in an electric field.
- Two types: positive and negative. Like charges repel; unlike charges attract (Fundamental Law of Electrostatics).
- Conservation of charge: Total charge in an isolated system is constant.
- Quantization: Charge occurs in integer multiples of the elementary charge $e$: $q = n\cdot e$, where $e = 1.60\times 10^{-19}\ \mathrm{C}$ and $n\in\mathbb{Z}$.
- Subatomic carriers: electron $q_e = -1.60\times 10^{-19}\ \mathrm{C}$, proton $q_p = +1.60\times 10^{-19}\ \mathrm{C}$.
Units and prefixes
Definition: The SI unit of charge is the coulomb (C).
- $1\ \mu\mathrm{C} = 10^{-6}\ \mathrm{C}$, $1\ \mathrm{nC} = 10^{-9}\ \mathrm{C}$, $1\ \mathrm{pC} = 10^{-12}\ \mathrm{C}$.
- One coulomb equals approximately $6.25\times 10^{18}$ electrons.
Coulomb's law
Definition: Coulomb's law gives the magnitude of the electrostatic force between two point charges.
$$F = k\frac{|q_1 q_2|}{r^2}$$
where $k = 8.99\times 10^9\ \mathrm{N\cdot m^2/C^2} = 1/(4\pi\varepsilon_0)$ and $\varepsilon_0 = 8.854\times 10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$.
- Use magnitudes in the formula and determine direction by attraction/repulsion rules.
- Superposition: net force is vector sum of pairwise forces.
Charging methods and triboelectric series
- Charging by friction: electron transfer when rubbing different materials.
- Charging by contact (conduction): touching transfers charge until equilibrium.
- Charging by induction: bring a charged object near a conductor, ground temporarily to remove like charges, then remove ground and the external object.
Examples (brief)
- Removing $16\times 10^6$ electrons gives $q = (16\times 10^6)(1.60\times 10^{-19}) = +2.56\times 10^{-12}\ \mathrm{C}$.
- Force between $-5\ \mu\mathrm{C}$ and $+3\ \mu\mathrm{C}$ separated by $2\ \mathrm{mm}$:
$$F = (9\times 10^9)\frac{(5\times 10^{-6})(3\times 10^{-6})}{(2\times 10^{-3})^2} = 3.38\times 10^4\ \mathrm{N}\quad\text{(attractive)}$$
Chapter 2: Electric Field and Dipoles
Electric field definition
Definition: The electric field $\mathbf{E}$ at a point is the force per unit positive test charge: $\mathbf{E} = \mathbf{F}_e/q_0$, units N/C or V/m.
- A positive test charge experiences force in direction of $\mathbf{E}$; a negative test charge experiences the opposite.
Field of a point charge
Definition: Electric field of a point charge $q$ at distance $r$ is
$$\mathbf{E} = k\frac{q}{r^2}\hat{r}$$
- For $q>0$ the field points radially outward; for $q<0$ it points radially inward.
- Superposition applies: $\mathbf{E}_{\text{net}} = \sum \mathbf{E}_i$.
Field lines and properties
- Field lines originate on positive charges and terminate on negative charges.
- Density of lines indicates field strength; lines never cross.
Electric dipoles
Definition: A dipole consists of charges $+q$ and $-q$ separated by distance $d$. The dipole moment is $\mathbf{p} = q\mathbf{d}$ pointing from $-q$ to $+q$.
- Torque in uniform field: $\mathbf{\tau} = \mathbf{p}\times\mathbf{E}$, magnitude $\tau = pE\sin\phi$.
- Potential energy: $U = -\mathbf{p}\cdot\mathbf{E} = -pE\cos\phi$.
Example: A dipole with $q=2.0\times 10^{-19}\ \mathrm{C}$ and $d=0.5\ \mathrm{nm}$ in $E=5.0\times 10^6\ \ma
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Electrostatics Essentials
Klíčové pojmy: Charge is quantized: $q = n\cdot e$ with $e=1.60\times 10^{-19}\ \mathrm{C}$, Coulomb's law: $F = k\dfrac{|q_1 q_2|}{r^2}$ and use attraction/repulsion for direction, Electric field: $\mathbf{E}=\mathbf{F}_e/q_0$, point charge field $\mathbf{E}=k\dfrac{q}{r^2}\hat{r}$, Superposition principle: sum fields or forces vectorially, potentials scalarly, Electric potential of point charge: $V = k\dfrac{q}{r}$ and $\mathbf{E}=-\nabla V$, Gauss's law: $\oint\mathbf{E}\cdot d\mathbf{A}=Q_{\text{encl}}/\varepsilon_0$, use symmetry, Dipole moment $\mathbf{p}=q\mathbf{d}$, torque $\tau = pE\sin\phi$, potential energy $U=-pE\cos\phi$, Capacitance $C=Q/V$, parallel-plate $C=\varepsilon_0 A/d$, energy $U=\tfrac{1}{2}CV^2$, Infinite line field: $E=\dfrac{\lambda}{2\pi\varepsilon_0 r}$; infinite plane: $E=\dfrac{\sigma}{2\varepsilon_0}$, Inside conductor in electrostatic equilibrium $E=0$ and excess charge resides on surface