Electricity and Magnetism B-Fields from Moving Charges

Electricity and Magnetism B-Fields from Moving Charges Lana Sheridan De Anza College Feb 28, 2018

Last time force on a curved current carrying wire torque on a wire loop magnetic dipole moment

Overview B-field from a moving charge fields from currents

Magnetic fields from moving charges and currents We are now moving into chapter 30. Anything with a magnet moment creates a magnetic field. This is a logical consequence of Newton s third law.

Magnetic Permeability A constant we will need is: µ 0 = 4π 10 7 Tm/A µ 0 is called the magnetic permeability of free space. It arises when we look at magnetic fields because of our choice of SI units. Whenever we use µ 0 we assume we are considering the magnetic field to be in a vacuum or air. µ 0 is not the magnetic dipole moment µ! Another notation coincidence.

Magnetic fields from moving charges A moving charge will interact with other magnetic poles. This is because it has a magnetic field of its own. The field for a moving charge is given by the Biot-Savart Law: B = µ 0 q v ˆr 4π r 2

Magnetic fields from moving charges A moving charge will interact with other magnetic poles. This is because it has a magnetic field of its own. The field for a moving charge is given by the Biot-Savart Law: B = µ 0 q v ˆr 4π r 2 Quite similar to the E-field from a charge: E = 1 4πɛ 0 q r 2 ˆr

Magnetic fields from moving charges B = µ 0 q v ˆr 4π r 2 1 Figure from rakeshkapoor.us.

Magnetic fields from currents Consider the field from an infinitesimal amount of charge, dq: db = µ 0 dq v ˆr 4π r 2 We can deduce from this what the magnetic field due to the current in a small piece of wire is. Current is made up of moving charges! dq v = dq ds dt = dq ds = I ds dt We can replace dq v in the equation above.

Magnetic fields from currents This element of current creates a magnetic field at P, into the page. current-length element a differential magnetic nt P. The green (the ) at the dot for point P db : is directed into the i ids θ ds ˆ r r P Current distribution db(into page) This is another version of the Biot-Savart Law: db = µ 0 I ds ˆr 4π r 2 where db is the magnetic field from a small segment of wire, of length ds.

mmation Magnetic fields from currents The magnetic field vector ecause of at any point is tangent to s a scalar, Magnetic field a circle. around a wire segment, viewed end-on: being the Wire with current into the page a current- (29-1) hat points constant, B B (29-2) the cross Fig. 29-2 The magnetic field lines produced by a current in a long straight wire

Magnetic fields from currents IC FIELDS How to determine DUE TO CURRENTS the direction of the field lines (right-hand rule): di- - -2, at - he e - ge, (a) B i (b) Here is a simple right-hand rule for finding the direction of the set up by a current-length element, such as a section of a long wire: B i The thum current's The fing the field direction tangent

trary shape carrying a current i.we want to find point Using P.We the first Biot-Savart mentally divide the Law wire into define for each element a length vector db = µ ds : that on is the direction of the current in ds.we 0 I ds can ˆr t-length element to be i ds : ;we wish to 4π calculate r 2 ypical current-length element. From experiment electric Tofields, calculate can be a superimposed B-field vector, to we find will a net need to integrate over s. net field B : at P by summing, via integration, the This is a path integral, similar to what we needed to evaluate: This element of current V = creates E a ds magnetic field at P, into the page. nt P i ids θ ds ˆ r r P Current distribution db(into page) Now, however, we are finding the cross product: the perpendicular component of s relative to the vector r.

Line Integrals There is one other special piece of notation used with some line integrals: This symbol means that the integral starts and ends at the same point. The path is a loop. (We also used the notation before in Gauss s law to indicate a closed surface.)

Magnetic field from a long straight wire The Biot-Savart Law, db = µ 0 I ds ˆr 4π r 2 implies what the magnetic field is at a perpendicular distance a from an infinitely long straight wire: B = µ 0I 2πa

B, and C in terms of magnitude of the n Magnetic just the length fieldelement from da S slong shown straight wire Add up the B-field contributed by each infinitesimal length of wire carrying current I. db = µ 0 I ds ˆr y 4π r S d s 2 dx P Following from the diagram: u r a ds ˆr = cos θ dx k = a r dx k nductor rˆ d S s x O I x B = µ 0 4π I a (x 2 + a 2 dx k ) 3/2 a y

B, and C in terms of magnitude of the n Magnetic just the length fieldelement from da S slong shown straight wire Add up the B-field contributed by each infinitesimal length of wire carrying current I. nductor y S d s dx P B = µ 0 4π I rˆ a d S s r x u y a O I x a (x 2 + a 2 dx k ) 3/2 = µ [ 0 4π I x a x 2 + a 2 = µ 0 4π I [ l ] l/2 l/2 a (l/2) 2 + a 2 ] k k

Magnetic field from a long straight wire Now consider a very long wire, l. [ ] µ 0 B = lim l 4π I l agnetic field lines due to an infinite wire carrying a current k (l/2) 2 I. + a 2 f the magnetic field due to a long = µ 0I straight 2 4π a k wire can be determined (Figure 9.1.5). In general, the field will be a tangential vector: B = µ 0I 2πr ˆϕ at a distance r from the center of the wire. irection of 1 Figure the magnetic from MIT s field Visualizing due to an EM, infinite Liao, straight Dourmashkin, wire and Belcher.

Summary field from a moving charge field from a current 3rd Test Friday, March 9. Homework Collected homework 3, posted online, due on Monday, Mar 5. Serway & Jewett: PREVIOUS: Ch 29, Problems: 47, 51, 57 PREVIOUS: Ch 30, Problems: 49 NEW: Ch 30, Problems: 3, 5, 9, 13, 19