Himanish Agarwal ❚
Notes on software, systems, and the science of making machines learn

Electromagnetic Theory

[Wed Oct 20, 2021]
— Himanish

Vector Analysis #

Vector Algebra #

  • \[\mathbf{B} \times \mathbf{A} = - (\mathbf{A} \times \mathbf{B})\]
  • \[(\textbf{A} \times \textbf{B}) \cdot (\textbf{C} \times \textbf{D}) = (\textbf{A} \cdot \textbf{C})(\textbf{B} \cdot \textbf{D}) - (\textbf{A} \cdot \textbf{D})(\textbf{B} \cdot \textbf{C}) \]

Differential Calculus #

  • \[ \nabla (\textbf{A} \cdot \textbf{B}) = \textbf{A} \times (\nabla \times \textbf{B}) + \textbf{B} \times (\nabla \times \textbf{A}) + (\textbf{A} \cdot \nabla)\textbf{B} + (\textbf{B} \cdot \nabla)\textbf{A}\]
    • \[\nabla \cdot (f\textbf{A}) = f(\nabla \cdot \textbf{A}) + \textbf{A} \cdot(\nabla f) \]
  • \[\nabla \times(f \mathbf{A})=f(\nabla \times \mathbf{A})-\mathbf{A} \times(\nabla f) \]
  • \[\boldsymbol{\nabla} \times(\mathbf{A} \times \mathbf{B})=(\mathbf{B} \cdot \boldsymbol{\nabla}) \mathbf{A}-(\mathbf{A} \cdot \boldsymbol{\nabla}) \mathbf{B}+\mathbf{A}(\boldsymbol{\nabla} \cdot \mathbf{B})-\mathbf{B}(\mathbf{\nabla} \cdot \mathbf{A})\]
  • \[\nabla\left(\frac{f}{g}\right) =\frac{g \nabla f-f \nabla g}{g^{2}}\]

\[\nabla \cdot\left(\frac{\mathbf{A}}{g}\right) =\frac{g(\nabla \cdot \mathbf{A})-\mathbf{A} \cdot(\nabla g)}{g^{2}} \] \[\nabla \times\left(\frac{\mathbf{A}}{g}\right) =\frac{g(\nabla \times \mathbf{A})+\mathbf{A} \times(\nabla g)}{g^{2}}\]

Integral Calculus #

  • Fundamental Theorem for Gradients

    \[ \int_a^b(\nabla T) \cdot d \textbf{l} = T(\textbf{b}) - T(\textbf{a}) \]

  • Fundamental Theorem for Divergences (Gauss/Green)

    \[\int_{\mathcal{V}}(\nabla \cdot \mathbf{v}) d \tau=\oint_{\mathcal{S}} \mathbf{v} \cdot d \mathbf{a}\]

  • Fundamental Theorem for Curls (Stokes)

    \[\int_{\mathcal{S}}(\nabla \times \mathbf{v}) \cdot d \mathbf{a}=\oint_{\mathcal{P}} \mathbf{v} \cdot d \mathbf{l}\]

Curvilinear Coordinates #

Spherical Coordinates #

  1. \[ d\tau = r^2\sin \theta \ dr \ d\theta \ d\phi \]
  2. On the surface of a sphere, r is constant, thus\[ d \boldsymbol{a_1} = r^2\sin \theta \ d \theta \ d\phi \ \hat{r}\]
  3. On the \(xy\) plane, \(\theta\) is constant, thus \[d \boldsymbol{a_2} = r \ dr \ d\phi \ \hat{\theta}\]
  4. \[ \hat{r} = \sin θ \cos φ \ \boldsymbol{\hat{x}} + \sin θ \sin φ \ \boldsymbol{\hat{y}} + \cos θ \ \boldsymbol{\hat{z}}\]
  5. \[x = r\sin \theta \cos \phi, \quad y = r\sin \theta \sin \phi, \quad z = r \cos \theta\]

Cylindrical Coordinates #

Dirac Delta Function #

  1. \[\boxed{\int_{-\infty}^{\infty} f(x) \delta(x-a) d x=f(a)}\]
  2. \[\nabla \cdot\left(\frac{\boldsymbol{\hat{r}}}{r^{2}}\right)=4 \pi \delta^{3}(\boldsymbol{r})\]

Helmholtz Theorem #

\[\textbf{F} = −∇V + ∇ × \textbf{A}\]

Electrostatics #

Electric Field #

Coloumb’s Law #

Force on a test charge Q due to a single point charge q, that is at rest a distance \(*r*\) away \[F = \frac{1}{4\pi \epsilon_0} \frac{qQ}{r^2} \boldsymbol{\hat{r}}\] where \(\boldsymbol{r} = \textbf{r} - \textbf{r’}\) is the vector from source charge to field point.

Electric Field #

\[\textbf{F}=Q \textbf{E}\] where \[\textbf{E}(\textbf{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{r_i^2}\boldsymbol{\hat{r_i}}\]

Continuous Charge Distributions #

\[\textbf{E}(\textbf{r}) = \frac{1}{4\pi\epsilon_0} \int\frac{1}{r_i^2}\boldsymbol{\hat{r_i}}dq\] \[= \frac{1}{4\pi\epsilon_0} \int\frac{\lambda(\textbf{r’})}{r_i^2}\boldsymbol{\hat{r_i}}dl’\] \[= \frac{1}{4\pi\epsilon_0} \int\frac{\sigma(\textbf{r’})}{r_i^2}\boldsymbol{\hat{r_i}}da’\] \[= \frac{1}{4\pi\epsilon_0} \int\frac{\rho(\textbf{r’})}{r_i^2}\boldsymbol{\hat{r_i}}d\tau’\]

Div and Curl of E-Fields #

Field Lines, Flux, Gauss’ Law #

  • Flux is proportional to the number of field lines, as field strength is proportional to field line density (per unit area).

\[\phi_E = \oint(\textbf{E} \cdot d\mathbf{a})= \frac{Q_{enc}}{\epsilon_0}\] [more natural to use] \[\nabla\cdot\textbf{E} = \frac{\rho}{\epsilon_0}\] [tidier]

Application #

Symmetry is crucial to application of Gauss’s law. Only three kinds of symmetry work:

  1. Spherical symmetry. Make your Gaussian surface a concentric sphere.
  2. Cylindrical symmetry. Make your Gaussian surface a coaxial cylinder.
  3. Plane symmetry. Use a Gaussian “pillbox” that straddles the surface.

Curl of E #

\[(\boldsymbol{\nabla} \times \textbf{E}) = 0\]

Electric Potential #

One function that contains all the information of three functions, the components of \(\textbf{E}\) which are not independent due to zero curl, as \[\textbf{E} = -\nabla V\]

  • For convenience, we set \( V(\infty) = 0\), but this convention fails when the charge distribution itself extends to infinity. The remedy is simply to choose some other reference point, e.g. a point on an infinite plane

Poisson’s Equation #

From Gauss’ Law: \[\nabla^2V = -\frac{\rho}{\epsilon_0}\]

Laplace’s Equation #

\[[\rho = 0] \ \nabla^2V = 0 \]

Boundary Conditions #

  • Unless the symmetry of the problem allows a solution by Gauss’s law, it is generally to your advantage to calculate the potential first, as an intermediate step
  • The electric field always undergoes a discontinuity when you cross a surface charge σ. The normal component of \(\textbf{E}\) is discontinuous by an amount \(\frac{\sigma}{\epsilon_0}\)at any boundary. The tangential component of \(\textbf{E}\), by contrast, is always continuous.
  • The potential, meanwhile, is continuous across any boundary. However, the gradient of V inherits the discontinuity in E.

Work and Energy #

\[W = \frac{1}{2}\sum_{i=1}^n q_iV(\boldsymbol{r_i}) \] \[\]