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{2/1} = -F{1/2} = \frac{1}{4\pi\epsilon0} \frac{q1 q2}{r^2} u{12}
    • Where:
      • F_{2/1} is the force exerted on charge 2 by charge 1.
      • q1 and q2 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\epsilon0} \frac{q1}{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\epsilon0} \iiintV \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\epsilon0} \iintS \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\epsilon0} \intC \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{\Pi}(P), then E(P') = sym{\Pi} (E(P)).
    • If \Pi is a plane of antisymmetry and P' = sym{\Pi}(P), then E(P') = -sym{\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{K: M1 \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(M2) - V(M1) = -\int{M1}^{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 q1 and q2 separated by a distance r12, the potential energy is: Ep = \frac{q1 q2}{4\pi\epsilon0 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, Ep = \frac{1}{2} \sum{i=1}^{n} qi Vi, where Vi is the potential felt by charge qi.

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:
  • \oiintS E \cdot dS = \frac{q{int}}{\epsilon0}, where q{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\epsilon0} er
  • r ≥ a (outside the sphere):
    • E = \frac{\rho a^3}{3\epsilon0 r^2} er
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\epsilon0 r^2} = \frac{p cos(\theta)}{4\pi\epsilon0 r^2}, where
      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\epsilon0 r^3} (2cos(\theta)er + 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: Ep = -p \cdot E{ext}
  • Equilibrium positions: E_p is extremum when p and E are aligned.
    • Stable equilibrium: p and E{ext} are in the same direction (minimum Ep).
    • Unstable equilibrium: p and E{ext} are in opposite directions (maximum Ep).
  • 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
    • \iiintV \nabla \cdot E d\tau = \oiintS E \cdot dS = \frac{q}{\epsilon_0} (Gauss's theorem)
    • \nabla \cdot E = -\nabla^2 V = \frac{\rho}{\epsilon0} (Poisson's equation: \nabla^2 V + \frac{\rho}{\epsilon0} = 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: E2 - E1 = \frac{\sigma}{\epsilon0} n{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{ext} = \frac{\sigma}{\epsilon0} n_{ext}, where
    _{ext} is the outward normal.

VIII.2. PRESSION ELECTROSTATIQUE

  • The electrostatic pressure on the surface of a conductor is: p = \frac{1}{2} \epsilon0 E^2 = \frac{\sigma^2}{2 \epsilon0}.
  • 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}{V1 - V2} (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{\epsilonr \epsilon0 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