Using this expansion it is remarkably simple to find the electrostatic potential in all space, then via the negative gradient it follows directly the electric field.

Additionally, we discuss the behavior of unit vectors in curvilinear coordinates.

In this session, we explore the profound relationship between Electromagnetism and the theory of Special Relativity. By analyzing the invariance of Maxwell's equations under coordinate transformations, we derive the Lorentz factor ($\gamma$) and the transformations for space and time. The lecture covers inertial systems, the application of the chain rule for primed coordinates, and the formal connection between Galilean and Lorentz physics.

Since the system exhibits azimuthal symmetry, the potential reduces to $\Phi=\Phi(r, \theta)$, effectively reducing the 3D Laplace equation to a 2D problem. The system is subject to the non-homogeneous Dirichlet boundary condition $\Phi(r_0, \theta) = \sin^2 \theta$. The lecture covers the separation of variables method, the derivation of the Legendre and Cauchy-Euler equations, and the application of boundary conditions via polynomial expansion.

Technical Note / Errata: Please note that in the equations shown at 11:01 and 11:05, a $+$ sign is missing between the terms. Ensure you include it when following the derivation in your notes.

In this lecture, we establish the foundations of electrostatics by starting with Coulomb's experimental law and formally defining the electric field. We generalize the field for continuous charge distributions (linear, surface, and volume) to set up a robust mathematical framework. Using vector calculus, specifically by analyzing the curl of the generalized electric field ($\vec{\nabla} \times \vec{E}$), we mathematically prove that every electrostatic field is conservative. This proof justifies the existence of the scalar electric potential $\Phi$ and naturally leads to the derivation of the Poisson equation, marking the historical shift from thinking in terms of forces to thinking in terms of potentials.

Technical Note: In the field integral expressions shown around 10:00, the $4\pi$ factor was unintentionally omitted. However, it's worth noting that these specific expressions are entirely correct when working in Gaussian units (CGS), a system widely preferred by theoretical physicists.

Because the cylinder is infinite, the system exhibits translational symmetry along the z-axis, simplifying the potential to $\Phi=\Phi(\rho, \phi)$ and effectively reducing the 3D Laplace equation to a 2D problem in cylindrical coordinates. The interior is subject to the non-homogeneous Dirichlet boundary condition $\Phi(\rho_0, \phi) = \cos^2\phi$. The lecture covers the separation of variables method, solving the resulting radial and angular ordinary differential equations, and the application of boundary conditions via trigonometric expansion to determine the final series coefficients.

We calculate the electric field of a wire with uniform linear charge density $\lambda$ positioned along the z-axis. Rather than assuming symmetries outright, we let the vector calculus naturally reveal them. Formulating the position vectors in cylindrical coordinates, the integration leads to the cancellation of the z-component due to the parity of the function over a symmetric interval. The remaining radial component is solved via a trigonometric substitution, proving analytically that the field strictly points radially outward ($\hat{e_\rho}$) and exhibits cylindrical symmetry, which is the fundamental requisite for applying Gauss's Law.

We calculate the electric field of an infinite plane with a uniform surface charge density $\sigma$, specifically evaluating the field at an observation point on the z-axis. To facilitate the integration, we parameterize the source surface using polar coordinates. Through rigorous vector calculus, the integrations naturally reveal that the transverse components ($x$ and $y$) cancel out completely due to the parity of the functions. The remaining perpendicular $z$-component is solved analytically, demonstrating that the electric field magnitude is constant and independent of the distance from the plane. We also provide a physical argument for why this result extends to all space, setting the stage for a fully generalized mathematical proof in a future lecture.

In this lecture, we deduce Poisson's equation by uniting two fundamental principles of electrostatics: the conservative nature of the electric field ($\vec{\nabla} \times \vec{E} = \vec{0}$, implying $\vec{E} = -\vec{\nabla}\Phi$) and Gauss's Law. We explicitly demonstrate the mathematical transition from the integral form of Gauss's Law to its local differential form ($\vec{\nabla} \cdot \vec{E} = \rho_v/\varepsilon_0$) using the Divergence Theorem. By combining these relations, we obtain Poisson's equation ($\nabla^2\Phi = -\rho_v/\varepsilon_0$). Finally, we examine how this expression simplifies into Laplace's equation ($\nabla^2\Phi = 0$) for regions completely devoid of free charge.

In this video, we explore the fundamental principle of the conservation of electric charge, mathematically encapsulated by the continuity equation ($\vec{\nabla} \cdot \vec{J} + \frac{\partial \rho_v}{\partial t} = 0$). The lecture begins by building an intuitive, visual understanding of how current density relates to the time evolution of charge distribution. Then, we perform a step-by-step mathematical derivation starting from Maxwell's equations, demonstrating how the core laws of electrodynamics inherently demand the absolute conservation of charge.

In standard electrostatics problems, the electric force is typically calculated acting upon a simple point-like test charge. In this lecture, we elevate the complexity by analyzing the force exerted on a continuous charge distribution, specifically, a finite wire interacting with the field of a parallel infinite wire. We discuss the physical setup and mathematically transition from the discrete force equation to a continuous vector integral ($d\vec{F} = \vec{E} dq$). We explicitly demonstrate how to set up the limits and evaluate the integral over the geometry of the finite test wire, concluding with a practical application of the derived analytical formula.

In this lecture, we calculate the magnetic field $\vec{B}$ produced by an infinite wire with steady current $I$. By parameterizing the source in cylindrical coordinates along the z-axis, the Biot-Savart integral is set up. We explicitly perform the vector cross-product between the current element $d\vec{l}'$ and the separation vector, demonstrating analytically why the resulting field is purely azimuthal ($\hat{e_\phi}$).