8: On the Electrodynamics of Moving Bodies
- Page ID
- 5465
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 8.1: Introduction to Electrodynamics of Moving Bodies
- This page introduces the motion of charged particles in electric and magnetic fields, inspired by Einstein's Special Theory of Relativity, though it focuses on nonrelativistic speeds. It is divided into sections addressing the motion in electric fields, magnetic fields, and their combinations, progressing in complexity. Some sections may present challenges and can be skipped in an initial reading for less experienced readers.
- 8.2: Charged Particle in an Electric Field
- This page discusses the behavior of a charged particle in an electric field, highlighting how it accelerates and the relationship between kinetic and potential energy as it moves through voltage. It includes the formula for velocity and notes that at high voltages, relativistic effects must be considered. Furthermore, it describes the parabolic trajectory of particles moving at an angle in an electric field.
- 8.3: Charged Particle in a Magnetic Field
- This page explores the interaction between electric currents and magnetic fields, focusing on the Lorentz force acting on charged particles in motion. It presents the mathematical relationship illustrating that the force is perpendicular to both particle velocity and the magnetic field. Circular and helical motion of particles is examined, alongside concepts like cyclotron angular speed.
- 8.4: Charged Particle in an Electric and a Magnetic Field
- This page discusses the motion of a charged particle in perpendicular uniform electric and magnetic fields, using Lorentz force equations. It describes how the particle undergoes circular motion aligned with the magnetic field while drifting in the electric field's direction, resulting in cycloidal paths. The page highlights the complexity of the trajectory, influenced by varying initial conditions, and presents parametric equations to illustrate the motion.
- 8.5: Motion in a Nonuniform Magnetic Field
- This page covers the motion of an electron in a magnetic field generated by an electric current in a wire, emphasizing the Lorentz force and equations of motion in cylindrical coordinates. It discusses various initial conditions affecting the electron's trajectory, introducing concepts like perineme and aponeme. The electron exhibits a helical path influenced by conservation laws, and the roles of initial velocity components and pitch angle are highlighted.
- 8.6: Appendix. Integration of the Equations
- This page covers numerical integration methods for equations with singularities at the perineme and aponeme. It presents substitutions to handle these singularities, allowing for power series expansions and term-by-term integration. Specific integral results are provided for certain intervals, expressed as series in terms of a small parameter \(\epsilon\), alongside constants that define these expansions and their relationships, aiding in integral calculations.
Thumbnail: Trajectory of a particle with a positive ornegative charge q under the influenceof a magnetic field B, which is directed perpendicularly out of the screen. (CC BY-SA; Jeffrey W. Schnick).