36 lines
1.3 KiB
Markdown
36 lines
1.3 KiB
Markdown
# EEMAGS Equation Sheet
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## Curl and Divergence Identities
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1. The Laplacian: can operate on a scalar or vector field
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$$\nabla \cdot (\nabla f) = \nabla^2 f$$
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$$\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}$$
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2. The curl of a gradient is $0$
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$$\nabla \times (\nabla f) = 0$$
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3. The gradient of divergence is a scalar
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$$\nabla (\nabla \cdot \vec{f})$$
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4. The divergence of curl is $0$
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$$\nabla \cdot (\nabla \times \vec{v}) = 0$$
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5. Curl of curl
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$$\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$$
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## Maxwell's Equations
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Gauss's Law for $\vec{E}$-fields:
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$$\Phi_E =\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0}$$
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$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$
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Gauss's Law for $\vec{B}$-fields:
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$$\int \vec{B} \cdot d\vec{s} = 0 $$
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$$ \nabla \cdot \vec{B} = 0$$
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Faraday's Law
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$$\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} $$
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$$ \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}$$
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Ampere's Law
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$$\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) = \mu_0 I_{enc}$$
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$$ \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$
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## Work and Voltage
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$$\Delta V = -\int \vec{E} \cdot d\vec{l}$$
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$$-\nabla V = \vec{E}$$
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