Electromagnetism - Electrostatics

CH.26 : ELECTROSTATIQUE

• Deals with electric fields created by IMMOBILE charges.
• Next chapter: magnetostatics (magnetic fields from PERMANENT currents).
• Chapter 28: induction (time-varying fields) shows the link between electric and magnetic fields, introducing the concept of "electromagnetic field".

I. NOTION DE CHAMP ELECTROSTATIQUE

I.1. LOI DE COULOMB

• Describes electrostatic forces between two point charges.
• Equation: $F<em>{2/1} = -F</em>{1/2} = \frac{1}{4\pi\epsilon<em>0} \frac{q</em>1 q<em>2}{r^2} u</em>{12}$
  • Where:
    • $F_{2/1}$ is the force exerted on charge 2 by charge 1.
    • $q<em>1$ and $q</em>2$ are the magnitudes of the charges.
    • $r$ is the distance between the charges.
    • $u_{12}$ is the unit vector pointing from charge 1 to charge 2.
    • $\epsilon_0$ is the permittivity of free space.
• Like charges repel, opposite charges attract.

I.2. CHAMP CREE PAR UNE CHARGE PONCTUELLE

• Defines the electric field created by a point charge.
• The electric field $E$ at a point is the force per unit charge that a small positive test charge would experience at that point.
• Equation: $E(M) = \frac{F}{q} = \frac{1}{4\pi\epsilon<em>0} \frac{q</em>1}{r^2} u_{12}$

I.3. CHAMP CREE PAR UNE DISTRIBUTION DE CHARGES

I.3.1. Echelle d’observation mésoscopique

• Microscopic scale: charges are discontinuous, and charge density $\rho$ varies significantly.
• Macroscopic scale: our scale of observation.
• Mesoscopic scale: intermediate scale, where volumes are around $1 \mu m^3$.
• At the mesoscopic scale, quantities are averaged over these volumes, resulting in smoother variations.

I.3.2. Distribution volumique de charges

• The electric field due to a volumetric charge distribution is given by:
• $E(M) = \frac{1}{4\pi\epsilon<em>0} \iiint</em>V \frac{\rho(P)}{r^2} u d\tau$
  • Where:
    • $\rho(P)$ is the volume charge density at point P.
    • $r$ is the distance between P and M.
    • $u$ is the unit vector pointing from P to M.
    • $d\tau$ is the infinitesimal volume element.

I.3.3. Modèles surfacique et linéique

• Surfacique (Surface Charge Density):
  • If one dimension is much smaller than the others, it's approximated as a charged surface.
  • The electric field is: $E(M) = \frac{1}{4\pi\epsilon<em>0} \iint</em>S \frac{\sigma(P)}{r^2} u dS$
    • Where: $\sigma = \frac{dq}{dS}$ (charge per unit area, in $C/m^2$).
• Linéique (Linear Charge Density):
  • For a filiform distribution, the electric field is: $E(M) = \frac{1}{4\pi\epsilon<em>0} \int</em>C \frac{\lambda(P)}{r^2} u dl$
    • Where: $\lambda = \frac{dq}{dl}$ (charge per unit length, in $C/m$).

II. INVARIANCES ET SYMETRIES

II.1. EXEMPLES D’INVARIANCES

• Rotation around an axis: In cylindrical or spherical coordinates, the field doesn't depend on the angle of rotation ($\theta$ or $\phi$).
• Translation along an axis: The field doesn't depend on the variable associated with that axis.

II.2. SYMETRIES ET ANTISYMETRIES

• Plane of Symmetry: A plane $\Pi$ is a plane of symmetry if for every point P in the distribution, its symmetric point P' has the same charge as P.
• Plane of Antisymmetry: A plane $\Pi$ is a plane of antisymmetry if for every point P in the distribution, its symmetric point P' has the opposite charge to that of P.

II.3. SITUATIONS A FORTE SYMETRIE

• Cylindrical Symmetry: If there's invariance by rotation and translation around an axis, quantities like the electric field or charge density depend only on $r$ (distance from the axis).
• Spherical Symmetry: If there's invariance by rotation according to $\theta$ and $\phi$ (in spherical coordinates), the same quantities depend only on $r$ (distance from the origin).
• Unidimensional Distribution: In Cartesian coordinates, if there's invariance by translation along two axes, quantities depend only on the variable associated with the third axis.

II.4. PRINCIPE DE CURIE

II.4.1. Enoncé du Principe

• "The symmetry of the effects is at least equal to that of the causes."
• The effects can be more symmetrical than the causes.

II.4.2. Application aux grandeurs vectorielles

• For vector quantities whose direction doesn't depend on a convention of orientation of rotations in space (polar vectors):
  • If $\Pi$ is a plane of symmetry and $P' = sym<em>{\Pi}(P)$, then $E(P') = sym</em>{\Pi} (E(P))$.
  • If $\Pi$ is a plane of antisymmetry and $P' = sym<em>{\Pi}(P)$, then $E(P') = -sym</em>{\Pi} (E(P))$.
  • If $\Pi$ is a plane of symmetry passing through point M, then $E(M) \in \Pi$.
  • If $\Pi$ is a plane of antisymmetry passing through point M, then $E(M) \perp \Pi$.
• Vectors like velocity ($v$), acceleration ($a$), and force ($F = ma$) obey Curie's principle and are called "polar" or "true" vectors.
• Axial vectors (pseudo-vectors) like angular momentum ($\sigma = OM \land mv$) or magnetic field ($B$) are linked to a convention of orientation and follow different symmetry rules.

III. NOTION DE POTENTIEL ELECTROSTATIQUE

III.1. CIRCULATION DU CHAMP ELECTROSTATIQUE

III.1.1. Définition

• The circulation of the electric field $E$ along a curve (K) from point M1 to M2 is defined as:
  • $C = \int<em>{K: M</em>1 \rightarrow M_2} E \cdot dl$
• The circulation of a force represents its work.

III.1.2. Propriétés

• The circulation of $E$ between M1 and M2 is independent of the path taken; it's "conservative".
• The circulation over a closed curve (contour) is zero: $\oint_{(C)} E \cdot dl = 0$

III.2. INTRODUCTION DU POTENTIEL

III.2.1. Définition

• Define the potential difference between two points M1 and M2 as:
  • $V(M<em>2) - V(M</em>1) = -\int<em>{M</em>1}^{M_2} E \cdot dl$
  • Where $V$ is the electrostatic potential from which the field $E$ is derived.
• For infinitesimally close points, $dV = -E \cdot dl$
• Define the gradient operator (grad) such that $dV = grad(V) \cdot dl$.
• Therefore: $E = -grad(V)$
• This relationship is local and intrinsic.

III.2.2. Lien entre les surfaces équipotentielles et les lignes de champ

• For two neighboring points M and M' on the same equipotential surface, $dV = E \cdot dl = 0$.
• Therefore, $E \perp dl$ at every point M.
• The electric field is perpendicular to the equipotential surface.

III.2.3. Potentiel d’une distribution de charges

• Point Charge:
  • In spherical coordinates: $V(M) = \frac{q}{4\pi\epsilon_0 r}$, assuming $V(\infty) = 0$.
• Charge Distributions:
  • Volumetric: $V(M) = \frac{1}{4\pi\epsilon_0} \iiint \frac{\rho d\tau}{r}$
  • Surface: $V(M) = \frac{1}{4\pi\epsilon_0} \iint \frac{\sigma dS}{r}$
  • Linear: $V(M) = \frac{1}{4\pi\epsilon_0} \int \frac{\lambda dl}{r}$

IV. ENERGIE POTENTIELLE D’UNE CHARGE DANS UN CHAMP

IV.1. CHARGE DANS UN CHAMP ELECTROSTATIQUE « EXTERIEUR »

• For a charge q at point M in an electric field E, the work done by the electric force for a displacement $dl$ is: $\delta W = qE \cdot dl = -q grad(V) \cdot dl = -q dV$
• The potential energy is $E_p = qV(M)$.

IV.2. SYSTEME DE CHARGES EN INTERACTION

IV.2.1. Cas de 2 charges

• For two charges $q<em>1$ and $q</em>2$ separated by a distance r12, the potential energy is: $E<em>p = \frac{q</em>1 q<em>2}{4\pi\epsilon</em>0 r_{12}}$.

IV.2.2. Cas de n charges

• The total potential energy is: Ep = \sum{i<j} \frac{qi qj}{4\pi\epsilon0 r{ij}}.
• Or, $E<em>p = \frac{1}{2} \sum</em>{i=1}^{n} q<em>i V</em>i$, where $V<em>i$ is the potential felt by charge $q</em>i$.

V. THEOREME DE GAUSS

V.1. ENONCE DU THEOREME

• The flux of the electric field through a closed surface (S) is proportional to the enclosed charge:
• $\oiint<em>S E \cdot dS = \frac{q</em>{int}}{\epsilon<em>0}$, where $q</em>{int}$ is the total charge enclosed by the surface.

V.2. EXEMPLES D’APPLICATION DU THEOREME

V.2.1. Sphère chargée uniformément en volume

• Consider a sphere of radius a with uniform volume charge density $\rho$.
• r ≤ a (inside the sphere):
  • $E = \frac{\rho r}{3\epsilon<em>0} e</em>r$
• r ≥ a (outside the sphere):
  • $E = \frac{\rho a^3}{3\epsilon<em>0 r^2} e</em>r$

V.2.2. Plan uniformément chargé en surface

• Consider an infinite plane with uniform surface charge density $\sigma$.
• The electric field is perpendicular to the plane and has magnitude:
• $E = \frac{\sigma}{2\epsilon_0}$ (for z > 0).
• $E = -\frac{\sigma}{2\epsilon_0}$ (for z < 0).
• There's a discontinuity in the electric field at the surface: $E(0^+) - E(0^-) = \frac{\sigma}{\epsilon_0}$.

VI. NOTION DE DIPOLE ELECTROSTATIQUE

VI.1. CHAMP CREE PAR UN DIPOLE ELECTROSTATIQUE

VI.1.1. Définition d’un dipôle électrostatique

• An electric dipole consists of two point charges of opposite sign, separated by a distance much smaller than the distance at which the field is calculated.
• The dipole moment is defined as: $p = q \cdot AB = qae_z$
• Dipoles are important in chemistry for studying interactions between molecules.
• A "rigid dipole" has constant $a$ and $q$, while an "induced dipole" has varying values.

VI.1.2. Expression du potentiel créé

• The potential created by a dipole is:
  • $V(M) = \frac{p \cdot OM}{4\pi\epsilon<em>0 r^2} = \frac{p cos(\theta)}{4\pi\epsilon</em>0 r^2}$, where $<br />$ is the distance from the dipole to the point M and theta is the angle between the dipole moment and OM vector.

VI.1.3. Expression du champ créé

• The electric field created by a dipole:
  • $E = \frac{p}{4\pi\epsilon<em>0 r^3} (2cos(\theta)e</em>r + sin(\theta)e_{\theta})$
• Intrinsic form: $E(M) = \frac{1}{4\pi\epsilon_0} \frac{3(p \cdot OM)OM - r^2 p}{r^5}$

VI.1.4. Equation des lignes de champ

• In polar coordinates (r,θ), the equation of field lines is given by: $r = K sin^2(\theta)$, where K is a constant.

VI.2. ACTION D’UN CHAMP ELECTROSTATIQUE UNIFORME SUR UN DIPOLE

• The net force on a dipole in a uniform electric field is zero.
• The resulting torque on the dipole is given by: $\Gamma = p \land E_{ext}$
• In a non-uniform field, there is a net force: $F = (p \cdot grad)E_{ext}$

VI.3. ENERGIE POTENTIELLE D’UN DIPOLE DANS UN CHAMP EXTERIEUR

• The potential energy of a dipole in an external field is: $E<em>p = -p \cdot E</em>{ext}$
• Equilibrium positions: $E_p$ is extremum when p and E are aligned.
  • Stable equilibrium: p and $E<em>{ext}$ are in the same direction (minimum $E</em>p$).
  • Unstable equilibrium: p and $E<em>{ext}$ are in opposite directions (maximum $E</em>p$).
• Multipole expansion can be used to approximate the electric potential of charge distributions as a series of terms with successively decreasing influence at large distances.

VII. FORMULATION LOCALE DE L’ELECTROSTATIQUE

VII.1. EQUATIONS DE MAXWELL DE L’ELECTROSTATIQUE

• In electrostatics, Maxwell's equations are:
  • $\nabla \cdot E = \frac{\rho}{\epsilon_0}$ (Maxwell-Gauss)
  • $\nabla \times E = 0$ (Maxwell-Faraday)

VII.2. LIEN AVEC LA FORMULATION INTEGREE

• Using vector analysis:
  • $\nabla \times E = 0 \Leftrightarrow E = -\nabla V$
  • $\iiint<em>V \nabla \cdot E d\tau = \oiint</em>S E \cdot dS = \frac{q}{\epsilon_0}$ (Gauss's theorem)
  • $\nabla \cdot E = -\nabla^2 V = \frac{\rho}{\epsilon<em>0}$ (Poisson's equation: $\nabla^2 V + \frac{\rho}{\epsilon</em>0} = 0$)

VII.3. RELATIONS DE PASSAGE POUR LE CHAMP ET LE POTENTIEL

• For a volumetric charge distribution, E and V are continuous.
• For a surface charge distribution $\sigma$, the tangential component of E is continuous, and there's a discontinuity in the normal component: $E<em>2 - E</em>1 = \frac{\sigma}{\epsilon<em>0} n</em>{12}$. V remains continuous.

VIII. CONDUCTEURS EN EQUILIBRE ELECTROSTATIQUE

VIII.1. PROPRIETES D’UN CONDUCTEUR A L’EQUILIBRE

VIII.1.1. Définitions

• Conductor: A material where charges can move under any small force.
• Electrostatic Equilibrium: Charges are fixed at the mesoscopic scale.

VIII.1.2. Propriétés

• Inside a conductor at equilibrium:
  • $E_{int} = 0$
  • $V_{int} = constant$
  • $\rho_{int} = 0$
• The electric field outside a conductor at equilibrium is normal to the surface.

VIII.1.3. Théorème de Coulomb

• The electric field near the surface of a conductor at equilibrium is: $E<em>{ext} = \frac{\sigma}{\epsilon</em>0} n_{ext}$, where $<br /> _{ext}$ is the outward normal.

VIII.2. PRESSION ELECTROSTATIQUE

• The electrostatic pressure on the surface of a conductor is: $p = \frac{1}{2} \epsilon<em>0 E^2 = \frac{\sigma^2}{2 \epsilon</em>0}$.
• This pressure is directed outwards and tends to pull the surface charges away.

VIII.3. CONDENSATEURS

VIII.3.1. Définition

• A capacitor is a system of two conductors, one hollow and surrounding the other, separated by a vacuum or dielectric.
• Capacitance is defined as: $C = \frac{Q}{V<em>1 - V</em>2}$ (Q is the charge on the internal armature, V1 and V2 refer to voltage on each surface).

VIII.3.2. Calculs de capacité

• Spherical Capacitor:
  • $C = \frac{4\pi\epsilon_0 ab}{b-a}$
• Cylindrical Capacitor:
  • $C = \frac{2\pi\epsilon_0 h}{Ln(\frac{b}{a})}$
• Parallel Plate Capacitor:
  • $C = \frac{\epsilon_0 S}{e}$, where S is the area of the plates and e is the separation.
  • With a dielectric: $C = \frac{\epsilon<em>r \epsilon</em>0 S}{e}$

VIII.3.3. Energie électrostatique d’un condensateur plan

• The energy stored in a capacitor is: $W = \frac{1}{2}CU^2$, where U is the voltage across the capacitor.
• For a parallel-plate capacitor, the volume energy density is: $w = \frac{1}{2} \epsilon_0 E^2$