Browse & Filter

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• Phase velocity of an electromagnetic wave:$$v= \frac{c}{\sqrt{\varepsilon \mu}},$$ where $$c= \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$$
• In a travelling electromagnetic wave:$$E\sqrt{\varepsilon \varepsilon_0} = H\sqrt{\mu \mu_0}.$$  
• Space density of the energy of an electromagnetic field:$$ \omega = \frac{\vec{E}.\vec{D}}{2}+\frac{\vec{B}.\vec{H}}{2}.$$
• Flow density of electromagnetic energy, the Poynting vector:$$\vec{S}=\vec{E}\times\vec{H}.$$

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Damped oscillation in a circuit $$q=q_m e^{-\beta t} \cos(\omega t+ \alpha),$$ where $$\omega= \sqrt{ \omega ^2_0 -\beta^2, \omega_0=\frac{1}{\sqrt{LC}}}, \beta=\frac{R}{2L}.$$

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• Faraday's law of electromagnetic induction:$$\mathscr{C}_i=-\frac{d\Phi}{di}\tag{1}$$
• In the case of a solenoid and doughnut coil:$$\Phi=N \Phi_{1}  \tag{2}$$ where N is the number of turns, is the magnetic flux through each turn.
• Inductance of a solenoid:$$L=\mu \mu_0 n^2 V   \tag{3}$$
• Maxwell's equations in differential form: $$\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, \vec{\nabla}.\vec{B}=0, \\ \vec{\nabla} \times \vec{H} = \vec{j} + \frac{\partial \vec{D}}{\partial t}, \vec{\nabla}.\vec{D}=\rho,$$ where $\vec{\nabla} \times \equiv$ rot (the rotor) and $\vec{\nabla}. \equiv$ div (the divergence).

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• Ohm's law for an inhomogeneous segment of a circuit:$$ I=\frac{V_{12}}{R}=\frac{\phi_{1}-\phi_{2} + \mathscr{E}_{12}}{R},$$  where $V_{12}$ is the voltage drop across the segment.
• Differential form of Ohm's law:$$\vec{j}=\sigma (\vec{E} + \vec{E}^\star),$$ where $\vec{E}^\star$ is the strength of a field produced by extraneous forces.
• Kirchhoff's laws (for an electric circuit): $$ \sum I_k=0, \sum I_k R_k= \sum \mathscr{E}_k.$$
• Power P of current and thermal power Q: $$P=VI=(\phi_1 - \phi_2 +\mathscr{E}_{12})I, Q=RI^2.$$

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• Capacitance of a parallel-plate capacitor:$$C=\varepsilon {\varepsilon}_0 \frac{S}{d}$$
• Interaction energy of a system of point charges:$$W=\frac{1}{2}\sum q_i \phi_{i}. $$
• Energy of a charged capacitor:$$W=\frac{qV}{2}=\frac{q^2}{2C}=\frac{CV^2}{2}$$

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• Strength and potential of the field of a point charge $q$:$$ \vec{E}=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r^3}\vec{r},   \phi=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r}.$$
• Relation between field strength and potential:$$\vec{E}=- \vec{\nabla}\phi. $$ i.e. field strength is equal to the anti gradient of the potential.
• Gauss's theorem and circulation of the vector $\vec{E}$:$$\oint \vec{E}.d\vec{S}=\frac{q}{{\varepsilon}_0}, \oint  \vec{E}.d\vec{r}=0.$$