Study guide: Electrostatics and Electric Dipoles

📋 Course Outline

1. Static Fields and Maxwell Equations
2. Point Charges and Charge Distributions
3. Electric Field Symmetry and Geometry
4. Electrostatic Potential and Poisson Equation
5. Gauss Law and Interface Conditions
6. Electric Dipole Fields and Torque

📖 1. Static Fields and Maxwell Equations

★ Must-know

📌 Stationary charge distributions produce time-independent electrostatic fields, whereas charges moving at a constant rate produce time-independent magnetostatic fields.

• In a medium with permittivity ε and permeability μ, the static Maxwell equations are div(εE)=ρ, div(B)=0, rot(E)=0, and rot(B/μ)=j.

Further detail

• The integral static Maxwell equations express electric flux as enclosed charge, magnetic flux as zero, electric-field circulation as zero, and circulation of B/μ as enclosed current.

💡 Memory Hook

Stationary charges create electrostatic fields; steady currents create magnetostatic fields.

📖 2. Point Charges and Charge Distributions

🔑 Key Concepts & Definitions

• Electric field: The force per unit charge experienced by a test charge, so E=F/q.

• Charge density: Charge density describes charge per unit volume, area, or length: volume density ρ, surface density σ, and line density λ; the corresponding total charges are integrals of these densities over the volume, surface, or line.

★ Must-know

📐 Formula - Coulomb’s law gives the field of a point charge as $\vec E=\frac{q_0}{4\pi\varepsilon r^2}\vec u_r$, where ε is the medium permittivity and uᵣ points from the source charge toward the field point.

• Like charges repel and unlike charges attract, and the force between two charges acts along the line joining them.

Further detail

• The field of several point charges is obtained by vectorially adding the field contributions of each charge.

• For a region containing one type of carrier with number density n and charge q, the volume charge density is ρ=nq.

📖 3. Electric Field Symmetry and Geometry

★ Must-know

📌 Electric-field lines are perpendicular to equipotential surfaces and point toward decreasing potential.

📌 For a charge distribution symmetric about a plane, the electric field lies in that plane at points on the plane; for an antisymmetric distribution, the field is perpendicular to the plane there.

Further detail

• Electric-field lines can cross only at point charges or where the electric field is zero.

📌 Translation invariance of a charge distribution makes the electric field independent of position along the invariant axis, and rotational invariance makes it independent of the rotation angle.

• For an infinite uniformly charged wall of thickness e and volume density ρ, the field inside varies linearly with distance x from the midplane, while outside it is constant; in the thin-wall limit σ=ρe.

📖 4. Electrostatic Potential and Poisson Equation

🔑 Key Concepts & Definitions

• Electrostatic potential difference: The potential difference V_BA between A and B is the work supplied to move a positive test charge from A to B divided by its charge, and it equals the negative line integral of E from A to B.

★ Must-know

📌 In electrostatics, the circulation of E between two fixed endpoints is independent of the path because the field has zero circulation around every closed loop.

📐 Formula - The electric field is the negative gradient of potential, $\vec E=-\nabla V$.

Further detail

📐 Formula - For a point charge q with zero potential chosen at infinity, $V(r)=\frac{q}{4\pi\varepsilon r}$.

• For continuous volume, surface, and line charge distributions, the potential is found by integrating the contribution proportional to ρdτ/r, σdS/r, or λdl/r, respectively, with factor 1/(4πε).

📐 Formula - The electrostatic potential satisfies Poisson’s equation $\Delta V=-\frac{\rho}{\varepsilon}$ and, in a region without volume charge, Laplace’s equation $\Delta V=0$.

📖 5. Gauss Law and Interface Conditions

★ Must-know

📐 Formula - Gauss’s law states that the electric flux through any closed surface is the enclosed charge divided by permittivity: $\oiint_S \vec E\cdot d\vec S=\frac{Q_{\mathrm{int}}}{\varepsilon}$.

📌 Across an interface, the tangential components of the electrostatic field are continuous: E₂t=E₁t.

📐 Formula - Across an interface carrying surface charge density σ, the normal displacement jump satisfies $\varepsilon_2 E_{2n}-\varepsilon_1 E_{1n}=\sigma$.

Further detail

📌 Gauss’s law is most useful for calculating the field when the charge distribution has a high degree of symmetry.

• The electrostatic potential remains continuous when crossing a charged surface, even though the electric field may be discontinuous there.

📖 6. Electric Dipole Fields and Torque

🔑 Key Concepts & Definitions

• Electric dipole: An electric dipole consists of two point charges of opposite sign, +q and −q, separated by a distance d.
• Dipole moment: The electric dipole moment is $\vec p=q\overrightarrow{BA}$ for charges −q at B and +q at A.

★ Must-know

📐 Formula - At distances r much greater than the separation d, the dipole potential is $V=\frac{p\cos\theta}{4\pi\varepsilon_0r^2}$.

📐 Formula - A dipole in a uniform electric field experiences a torque $\vec C=\vec p\times\vec E$ that tends to align the dipole moment with the field.

Further detail

📐 Formula - In the far-field dipole approximation, the radial and polar electric-field components are $E_r=\frac{2p\cos\theta}{4\pi\varepsilon_0r^3}$ and $E_\theta=\frac{p\sin\theta}{4\pi\varepsilon_0r^3}$.

💡 Memory Hook

Picture a dipole as two opposite charges joined by a short arrow; in a uniform field, the arrow turns to align with the field.