Download Magnetostatics: Understanding Static Magnetic Fields and DC Currents - Prof. Phillip Duxbu and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 22 Chapter 8 of PS, Chapter 5 of Griffiths A. Basics of Magnetostatics Magnetostatics is the study of static magnetic fields and the steady currents (DC) that generate them. The MKS unit of magnetic field is the Tesla (T) and often the magnetic field is given in Gauss (G), the CGS unit. The relation is 10, 000G = 1T , so small fields are often given in Gauss. In the early 1800’s several researchers noted that there is a force between two parallel wires that carry a steady current . If the wires are separated by distance d, |F | l = µ0 2πd i1i2 attractive for currents in the same direction (1) where µ0 = 4π × 10−7N/A2 is the permeability of vacuum. If the currents are in opposite directions the wires repel. This result can be compared with the force between two infinite parallel line charges, with charge densities λ1, λ2. The magnitude of the force between the line charges, when separated by d, is given by, |F | l = q2 l |Eλ1| = λ1λ2 2π0d repulsive for like charges (2) where we used the electric field near a line charge E(r) = λ/2π0r. Other than the direction of the force these two results are similar. Clearly, excess charge leads to electric fields and a DC leads to magnetic fields. The total force is a superposition of these two effects, so that electrostatics and magnetostatics don’t affect each other. Ampere and Oersted also noticed that magnets are affected by a DC current in a wire and Ampere found that, ~B(r) = µ0i 2πr φ̂ Magnetic field near a wire (3) This is not such a leap as we already know that the electric field due to a line charge is ~E(r) = λ 2π0r r̂ Electric field near a line charge (4) The most remarkable difference between these two results is the direction of the fields - The electric field diverges from the line charge and is curl free (~∇∧ ~E = 0) - The magnetic field forms circles around the steady current and is divergence free (~∇ · ~B = 0) Notice that the magnetic field is not moving even though the steady current is 1 flowing and even though we often talk about the circulation of the magnetic field about the current-The magnetic field is static. The direction of circulation is given by the right hand rule, applied to either the magnetic field or to the current. B. Ampere’s law and related problems We start with Section 4 of PS or Section 5.3.3 of Griffiths by writing down Ampere’s law. Ampere realised that his measurements for the magnetic field near a wire may be written in the form of a path integral. ∮ ~B · d~l = µ0i = µ0 ∫ ~j · d ~A Ampere ′s law (5) Here d~l is a small vector along the direction of the path. For example for a circle the unit vector is tangent to the circle. Now consider the magnetic field around a long straight wire, where the current, i, is in the k̂ direction, then we have, ∫ 2π 0 B(r)φ̂ · rdφφ̂ = 2πrB(r) = µ0i (6) Solving for B(r) yields, ~B(r) = µ0i 2πr φ̂ (7) Ampere’s law is correct only for DC currents. Later in the course we will add an additional term which enables us to use this equation even when there are time dependent fields - the Maxwell displacement current term. There is a set of important problems that can be solved analytically using the integral form of Ampere’s law. The most basic is the line charge case as illustrated above. Another simple case is a sheet of current. In that case, consider a sheet of current lying in the x-y place, with the current directed along the positive x-direction, so that the surface current density is Kî, where K is the current per unit length. If we draw an L × H rectangular amperian loop around the current sheet, with L the length of a section parallel to the y-axis and H a section parallel to the z-direction. 2LB = µ0LK; so that ~B = − 1 2 µ0Kĵ, z > 0; ~B = 1 2 µ0Kĵ, z < 0. (8) A practically important and closely related case is that of an infinite solenoid of radius R with its axis along the z-axis, where there are n “turns” per unit length. In that case, the 2
