6: The Magnetic Effect of an Electric Current
- Page ID
- 5451
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Introduction
- A bar magnet has some properties that are quite similar to those of an electric dipole. The region of space around a magnet within which it exerts its magic influence is called a magnetic field, and its geometry is rather similar to that of the electric field around an electric dipole – although its nature seems a little different, in that it interacts with iron filings and small bits of iron rather than with scraps of paper or pith-balls.
- 6.2: Definition of the Amp
- If there are two parallel wires each carrying a current in the same direction, the two wires will attract each other with a force that depends on the strength of the current in each, and the distance between the wires.
- 6.3: Definition of the Magnetic Field
- This page defines the magnetic field's magnitude and direction through its effects on electric currents, without mentioning magnets. It describes how currents in a wire experience a force in a magnetic field, which is perpendicular to the current's direction. The field's intensity (B) is defined as the maximum force per unit length on unit current, measured in teslas, and the relationship between force, current, and magnetic field is expressed in vector notation using the right-hand rule.
- 6.4: The Biot-Savart Law
- When we were calculating the electric field in the vicinity of various geometries of charged bodies, we started from Coulomb’s Law, which told us what the field was at a given distance from a point charge. Is there something similar in electromagnetism which tells us how the magnetic field varies with distance from an electric current? Indeed there is, and it is called the Biot-Savart Law.
- 6.5: Magnetic Field Near a Long, Straight, Current-carrying Conductor
- This page explains the calculation of the magnetic field at a point \(P\) near a current-carrying conductor. It details how to derive the magnetic field from a small segment of the conductor using the formula \(dB = \frac{\mu I\,dx \cos \theta }{4\pi r^2}\). The equation is simplified and integrated to find the total magnetic field, given by \(B=\frac{\mu I}{2\pi a}\), illustrating the importance of cylindrical symmetry in this scenario.
- 6.6: Field on the Axis and in the Plane of a Plane Circular Current-carrying Coil
- This page derives the magnetic field at point P from a charged coil, comparing it with that of a charged ring. It presents the axial magnetic field formula \(B=\dfrac{\mu I a^2}{2(a^2+x^2)^{3/2}}\), simplifying to \(B=\dfrac{\mu I}{2a}\) at the coil's center, where the field peaks and diminishes with distance. Additionally, it details the calculation of the magnetic field in the ring's plane, using numerical integration for accurate graphical representation.
- 6.7: Helmholtz Coils
- If the separation between two identical parallel plane coils is equal to the radius of one of the coils, the arrangement is known as “Helmholtz coils." They are of particular interest.
- 6.8: Field on the Axis of a Long Solenoid
- This page examines the magnetic field produced by a solenoid based on its radius, turns per unit length, and current. It provides a derivation for the magnetic field along the solenoid's axis from an elemental ring of wire, concluding that for an infinite solenoid, the field is \(B = \mu n I\). Additionally, it raises questions regarding the field's behavior away from the axis, suggesting more in-depth analysis will follow.
- 6.9: The Magnetic Field H
- This page explains the connection between the magnetic field \(B\) and the auxiliary magnetic field \(H\), defined as \(H = B/\mu\). It includes equations relating \(H\) to current and geometry, emphasizing its SI units of \(\text{A m}^{-1}\) and vector nature. The page also outlines the relationship \(\textbf{B} = \mu \textbf{H}\) and notes that \(B\) and \(H\) are parallel in isotropic media but can vary in direction in anisotropic media due to tensor behavior of permeability.
- 6.10: Flux
- This page covers the definition and calculation of electric and magnetic fluxes, including \(E\)-flux (\(\Phi_E\)), \(D\)-flux (\(\Phi_D\)), \(B\)-flux (\(\Phi_B\)), and \(H\)-flux (\(\Phi_H\)). It explains how to compute these fluxes using surface integrals, specifies SI units (with \(\Phi_B\) in weber), and includes a summary table for easy reference of the units and dimensions associated with each field and flux type.
- 6.11: Ampère’s Theorem
- Ampère’s Theorem is concerned with the magnetic field around a current-carrying conductor that will enable us to calculate the magnetic field in its vicinity. This is Ampère’s Theorem argues that the line integral of the field H around any closed path is equal to the current enclosed by that path.
- 6.12: Boundary Conditions
- This page explains how electric and magnetic fields behave at the boundaries between different media, highlighting that certain components (normal and tangential) of electric and magnetic fields are continuous, while others depend on the material properties (permittivity and permeability). Examples involving solenoids illustrate these continuity conditions for better understanding.