Displacement Current and Electromagnetic Waves

Formulas

Maxwell’s Equations

Gauss’ Law for E fields

\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_{0}} \]

Gauss’ Law for B fields

\[ \oint \vec{B} \cdot d\vec{A} = 0 \]

Faraday’s Law

\[ \oint \vec{E} \cdot d\vec{l} = - \frac{d}{dt} \int \vec{B} \cdot d\vec{A} \]

Ampere’s Law (not modified)

\[ \oint \vec{B} \cdot d\vec{l} = \mu_{0} I_{enclosed} + ??? \]

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Displacement Current

\[ I_{D} = \epsilon_{0} \frac{d \Phi_{E}}{dt} \]

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Modified Ampere’s Law

\[ \oint \vec{B} \cdot \vec{dl} = \mu_{0} ( I + I_{D} ) \]

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Wave Review

• Amplitude – \(A\)
• Wave Number – \(k = \frac{2 \pi}{\lambda}\)
• Wavelength – \(\lambda\)
• Angular Frequency – \(\omega = \frac{2 \pi}{T}\)
• Period – \(T\)
• Frequency – \(f = \frac{1}{T}\)
• Velocity – \(v = \lambda f = \frac{\omega}{k}\)

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Other

\[ B_{y} = \frac{k}{ \omega } E_{0} cos( kz - \omega t ) \]

\[ B_{0} = \frac{E_{0}}{c} \]

\[ c = 3 \times 10^{8} m/s \]