Rowan-Classes/5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.md

EEMAGS Equation Sheet

Curl and Divergence Identities

1. The Laplacian: can operate on a scalar or vector field

\nabla \cdot (\nabla f) = \nabla^2 f

\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}

2. The curl of a gradient is 0

\nabla \times (\nabla f) = 0

3. The gradient of divergence is a scalar

\nabla (\nabla \cdot \vec{f})

4. The divergence of curl is 0

\nabla \cdot (\nabla \times \vec{v}) = 0

5. Curl of curl

\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}

Maxwell's Equations

Gauss's Law for $\vec{E}$-fields:

\Phi_E =\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0}

\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}

Gauss's Law for $\vec{B}$-fields:

\int \vec{B} \cdot d\vec{s} = 0 

 \nabla \cdot \vec{B} = 0

Faraday's Law

\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} 

 \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}

Ampere's Law

\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) = \mu_0 I_{enc}

 \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}

Work and Voltage

\Delta V = -\int \vec{E} \cdot d\vec{l}

-\nabla V = \vec{E}