Electrostatics – Advanced University‑Level Study Notes
(Prepared for a MIT‑style Physics course – “Electrostatics” – Program: JEECUP)
Prerequisites:
• Classical mechanics (Newtonian dynamics)
• Vector calculus (divergence, curl, Green’s identities)
• Introductory electromagnetism (Maxwell’s equations, SI units)
• Basic complex analysis (for multipole expansions)
Notation:
• Greek letters: (\rho) (volume charge density), (\sigma) (surface charge density), (\lambda) (line charge density)
• (\varepsilon_0) – vacuum permittivity, (\varepsilon_r) – relative permittivity
• (\mathbf{E}) – electric field, (\mathbf{D}) – electric displacement, (\phi) – electrostatic potential
• (\nabla\cdot) – divergence, (\nabla\times) – curl, (\nabla^2) – Laplacian
Units: SI (volts, coulombs, meters, etc.).
1. Introduction & Fundamental Assumptions
Electrostatics studies electric fields generated by static charges – i.e., all charges are at rest and their positions do not change with time. In this regime:
- No magnetic fields (since (\nabla\times \mathbf{E} = 0)).
- Time‑independent fields: (\partial_t \mathbf{E} = 0).
- Linear, isotropic media unless stated otherwise.
The central goal is to determine (\mathbf{E}(\mathbf{r})) and (\phi(\mathbf{r})) given a charge distribution and boundary conditions.
2. Coulomb’s Law – From First Principles
2.1 Statement
For two point charges (q_1) and (q_2) separated by vector (\mathbf{r} = \mathbf{r}_2-\mathbf{r}_1),
[ \boxed{;\mathbf{F}_{12} = k_e \frac{q_1q_2}{r^{2}};\hat{\mathbf{r}};} ]
with (k_e = \frac{1}{4\pi\varepsilon_0}).
2.2 Derivation from Gauss’s Law
-
Gauss’s Law (integral form):
[ \oint_{\partial V}\mathbf{E}\cdot d\mathbf{a} = \frac{Q_{\text{enc}}}{\varepsilon_0} ]
-
Apply to a spherical Gaussian surface of radius (r) centered on a point charge (q):
[ \mathbf{E}(r), 4\pi r^{2} = \frac{q}{\varepsilon_0} ;;\Rightarrow;; \boxed{;\mathbf{E}(r) = \frac{q}{4\pi\varepsilon_0}\frac{\hat{\mathbf{r}}}{r^{2}};} ]
-
The force on (q_2) follows from (\mathbf{F} = q_2 \mathbf{E}).
2.3 Vector Form & Superposition
For an arbitrary (possibly continuous) charge distribution (\rho(\mathbf{r}')):
[ \boxed{; \mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}!\int!\frac{\rho(\mathbf{r}')(\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^{3}},d^{3}r' ;} ]
Superposition principle guarantees linearity: fields add.
3. Electric Field and Potential
3.1 Differential Form – Maxwell’s Equations
In electrostatics:
[ \nabla \times \mathbf{E} = 0, \qquad \nabla \cdot \mathbf