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6: Electromagnetism
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/06%3A_Electromagnetism
This chapter discusses two major effects that arise when electric and magnetic fields are changing in time: the “electromagnetic induction” of an additional electric field by changing magnetic field, ...This chapter discusses two major effects that arise when electric and magnetic fields are changing in time: the “electromagnetic induction” of an additional electric field by changing magnetic field, and the reciprocal effect of the “displacement currents”- actually, the induction of an additional magnetic field by changing electric field.
2.9: Variable Separation – Polar Coordinates
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/02%3A_Charges_and_Conductors/2.09%3A_Variable_Separation__Polar_Coordinates
15a $\ \left(E_{0}>0\right)$ , the surface charge is positive on the right-hand side of the cylinder and negative on its left-hand side, thus creating a field directed from the right to the left, whi...15a $\ \left(E_{0}>0\right)$ , the surface charge is positive on the right-hand side of the cylinder and negative on its left-hand side, thus creating a field directed from the right to the left, which exactly compensates the external field inside the conductor, where the net field is zero. (Please take one more look at the schematic Fig.
2.3: Exercise Problems
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/02%3A_Charges_and_Conductors/2.03%3A_Exercise_Problems
Calculate the mutual capacitance between the terminals of the semi-infinite lumped-capacitor circuit shown in the figure on the right, and the law of decay of the applied voltage along the system. Use...Calculate the mutual capacitance between the terminals of the semi-infinite lumped-capacitor circuit shown in the figure on the right, and the law of decay of the applied voltage along the system. Use the method of images to find the Green’s function of the system shown in the figure on the right, where the bulge on the conducting plane has the shape of a semi-sphere of radius $\ R$ .
10.5: Density Effects and the Cherenkov Radiation
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/10%3A_Radiation_by_Relativistic_Charges/10.05%3A_Coulomb_Losses
\[\ \phi_{\mathbf{k}, \omega}=\frac{1}{(2 \pi)^{3}} \frac{q \delta(\omega-\mathbf{k} \cdot \mathbf{u})}{\varepsilon(\omega)\left[k^{2}-\omega^{2} \varepsilon(\omega) \mu(\omega)\right]}, \quad \mathbf... $\ \phi_{\mathbf{k}, \omega}=\frac{1}{(2 \pi)^{3}} \frac{q \delta(\omega-\mathbf{k} \cdot \mathbf{u})}{\varepsilon(\omega)\left[k^{2}-\omega^{2} \varepsilon(\omega) \mu(\omega)\right]}, \quad \mathbf{A}_{\mathbf{k}, \omega}=\frac{1}{(2 \pi)^{3}} \frac{\mu(\omega) q \mathbf{u} \delta(\omega-\mathbf{k} \cdot \mathbf{u})}{\left[k^{2}-\omega^{2} \varepsilon(\omega) \mu(\omega)\right]} \equiv \varepsilon(\omega) \mu(\omega) \mathbf{u} \phi_{\mathbf{k}, \omega}.\tag{10.108}$
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The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californ...The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
2: Charges and Conductors
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/02%3A_Charges_and_Conductors
Such problems are also broadly used in other parts of electrodynamics (and indeed in other fields of physics as well), so that following tradition, I will use this chapter’s material as a playground f...Such problems are also broadly used in other parts of electrodynamics (and indeed in other fields of physics as well), so that following tradition, I will use this chapter’s material as a playground for a discussion of various methods of boundary problem solution, and the special functions most frequently encountered in the process.
Essential Graduate Physics - Classical Electrodynamics (Likharev)
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)
Thumbnail: Electric field lines due to a point charge in the vicinity of PEC regions (shaded) of various shapes. (CC BY SA 4.0; K. Kikkeri).
5.2: Vector Potential and the Ampère Law
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/05%3A_Magnetism/5.02%3A_Vector_Potential_and_the_Ampere_Law
where the operator $\ \nabla$ acts in the $\ \mathbf{r}$ -space. (This equality may be readily verified by its Cartesian components, noticing that the current density is a function of \(\ \mathbf{r...where the operator $\ \nabla$ acts in the $\ \mathbf{r}$ -space. (This equality may be readily verified by its Cartesian components, noticing that the current density is a function of $\ \mathbf{r}^{\prime}$ and hence its components are independent of $\ \mathbf{r}$ .) Plugging Eq. (26) into Eq. (14), and moving the operator $\ \nabla$ out of the integral over $\ \mathbf{r}^{\prime}$ , we see that the magnetic field may be represented as the curl of another vector field – the so-called…
10.1: Liénard-Wiechert Potentials
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/10%3A_Radiation_by_Relativistic_Charges/10.01%3A_Lienard-Wiechert_Potentials
However, the rate of radiation’s arrival at the observation point scales as $\ 1 / d t$ , so that due to the non-zero velocity $\ \mathbf{u}_{\mathrm{ret}}$ of the particle, this rate differs from ...However, the rate of radiation’s arrival at the observation point scales as $\ 1 / d t$ , so that due to the non-zero velocity $\ \mathbf{u}_{\mathrm{ret}}$ of the particle, this rate differs from the charge arrival rate by the factor of $\ d t_{\mathrm{ret}} / d t$ , given by Eq. (16). (If the particle moves toward the observation point, $\ (\boldsymbol{\beta} \cdot \mathbf{n})_{\mathrm{ret}}>0$ , as shown in Fig.
8.10: Exercise Problems
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/08%3A_Radiation_Scattering_Interference_and_Diffraction/8.10%3A_Exercise_Problems
9.6 below, but I hope that the reader knows that in the non-relativistic case ( $\ \nu<<c$ ) the above formula for $\ \omega_{\mathrm{c}}$ may be readily obtained by combining the 2 nd Newton law \(...9.6 below, but I hope that the reader knows that in the non-relativistic case ( $\ \nu<<c$ ) the above formula for $\ \omega_{\mathrm{c}}$ may be readily obtained by combining the 2 nd Newton law $\ m \nu_{\perp}^{2} / R=q \nu_{\perp} B$ for the circular motion of the particle under the effect of the magnetic component of the Lorentz force (5.10), and the geometric relation $\ \nu_{\perp}=R \omega_{\mathrm{c}}$ . (Here $\ \mathbf{v}_{\perp}$ is particle’s velocity in the plane normal to …
8.7: Geometrical Optics Placeholder
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Essential_Graduate_Physics_-_Classical_Electrodynamics_(Likharev)/08%3A_Radiation_Scattering_Interference_and_Diffraction/8.07%3A_Geometrical_Optics_Placeholder
This fact gives the strict foundation for the notion of the wave ray (or beam), as the line perpendicular to the local front of a quasi-plane wave. - the so-called lensmaker’s equation expressing the ...This fact gives the strict foundation for the notion of the wave ray (or beam), as the line perpendicular to the local front of a quasi-plane wave. - the so-called lensmaker’s equation expressing the focus length $\ f$ of a lens via the curvature radii of its spherical surfaces and the refraction index of the lens material,

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