Abstract
This chapter outlines the basic principles of the electromagnetic theory in vacuo. First, the extension of the Lagrangian formalism to functions that depend on more than one variable is tackled: this yields useful tools for the analysis of continuous media. Next, the Maxwell equations are introduced along with the derivation of the electric and magnetic potentials, and the concept of gauge transformation is illustrated. The second part of the chapter is devoted to the Helmholtz and wave equations, both in a finite and infinite domain. The chapter finally introduces the Lorentz force, that connects the electromagnetic field with the particles’ dynamics. The complements discuss some invariance properties of the Euler equations, derive the wave equations for the electric and magnetic field, and clarify some issues related to the boundary conditions in the application of the Green method to the boundary-value problem.
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Notes
- 1.
Also called D’Alembert equation in the homogeneous case.
- 2.
The units in (4.19, 4.23, 4.24) are: \([\mathbf{D}] ={\mathrm{C~m^{-2}}}\), \([\rho] = {\mathrm{C~m^{-3}}}\), \([\mathbf{H}] = {\mathrm{A~m^{-1}}}\), \([\mathbf{J}] ={\mathrm{C~s^{-1}~m^{-2} = A~m^{-2}}}\), \([\mathbf{B}] ={\mathrm{V~s~m^{-2} = Wb~m^{-2} = T}}\), \([\mathbf{E}] ={\mathrm{V~m^{-1}}}\), where “C”, “A”, “V”, “Wb”, and “T” stand for Coulomb, Ampere, Volt, Weber, and Tesla, respectively. The coefficients in (4.19, 4.23, 4.24) differ from those of [4] because of the different units adopted there. In turn, the units in (4.25) are \([\varepsilon_0] ={\mathrm{C~V^{-1}~m^{-1} = F~m^{-1}}}\), \([\mu_0] ={\mathrm{s^2~F^{-1}~m^{-1} = H~m^{-1}}}\), where “F” and “H” stand for Farad and Henry, respectively, and those in (4.26) are \([\varphi] = {\mathrm{V}}\), \([\mathbf{A}] ={\mathrm{V~s~m^{-1} = Wb~m^{-1} }}\).
- 3.
The minus sign in the definition of ϕ is used for consistency with the definition of the gravitational potential, where the force is opposite to the direction along which the potential grows.
- 4.
Expressions of ϕ and \(\mathbf{A}\) obtained from (4.58, 4.59) after replacing \(t - \vert \mathbf{r} - \mathbf{q} \vert /c\) with \(t + \vert \mathbf{r} - \mathbf{q} \vert / c\) are also solutions of the wave equations (4.35). This is due to the fact that the Helmholtz equation (4.43) can also be solved by using \(G^\ast\) instead of G, which in turn reflects the time reversibility of the wave equation. However, the form with \(t - \vert \mathbf{r} - \mathbf{q} \vert / c\) better represents the idea that an electromagnetic perturbation, that is present in \(\mathbf{r}\) at the time t, is produced by a source acting in \(\mathbf{q}\) at a time prior to t.
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Problems
Problems
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4.1
Solve the one-dimensional Poisson equation \({\rm d}^2 \varphi / {\rm d} x^2 = - \rho (x) / \varepsilon_0\), with ρ given, using the integration by parts to avoid a double integral. The solution is prescribed at x = a while the first derivative is prescribed at x = c.
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4.2
Let c = a in the solution of Prob. 4.1 and assume that the charge density ρ differs from zero only in a finite interval \(a\le x\le b\). Find the expression of ϕ for x > b when both the solution and the first derivative are prescribed at x = a.
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4.3
In Prob. 4.2 replace the charge density ρ with a different one, say, . Discuss the conditions that leave the solution unchanged.
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4.4
In Prob. 4.2 remove the charge density ρ and modify the boundary conditions at a so that the solution for x > b is left unchanged.
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4.5
Using the results of Probs. 4.2 and 4.3, and assuming that both M 0 and M 1 are different from zero, replace the ratio \(\rho / \varepsilon_0\) with \(\mu \, \delta ( x - h )\) and find the parameters μ, h that leave M 0, M 1 unchanged. Noting that h does not necessarily belong to the interval \([a,b]\), discuss the outcome for different positions of h with respect to a.
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Rudan, M. (2015). Electromagnetism. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_4
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_4
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