Physics 2113 Jonathan Dowling Lecture 28: MON 23 MAR Magnetic Fields Due to Currents: Biot-Savart Law Jean-Baptiste Felix Savart Biot (1774-1862) (17911841) What Are We Going to Learn? A Road Map Electric charge Electric force on other electric charges Electric field, and electric potential Moving electric charges : current Electronic circuit components: batteries, resistors, capacitors

Electric currents Magnetic field Magnetic force on moving charges Time-varying magnetic field Electric Field More circuit components: inductors. Electromagnetic waves light waves Geometrical Optics (light rays). Physical optics (light waves) Electric Current: A Source of Magnetic Field Oresteds Observation : an electric current creates a magnetic field Simple experiment: hold a current-carrying wire near a compass needle! I

B Wire with current INTO page B B Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand. While performing his electric

demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon. New Right Hand Rule! Point your thumb along the direction of the current in a straight wire i

The magnetic field created by the current consists of circular loops directed along B your curled fingers. The magnetic field gets weaker with distance: For long wire its a 1/R Law! You can apply this to ANY straight wire (even a small differential element!)

What if you have a curved wire? Break into small elements. Direction of B! i 29.2: Magnetic Field due to a Long Straight Wire: Fig. 29-4 A right-hand rule gives the direction of the magnetic field due to a current in a wire. (a) The magnetic field B at any point to the left of the wire is perpendicular to the dashed radial line and directed into the page, in the direction of the fingertips, as indicated by the x. (b) If the current is reversed, at any point to the left is still perpendicular to the dashed radial line but now is directed out of the page, as indicated by the dot. Superposition: ICPP Magnetic fields (like electric I-OUT

fields) can be superimposed -- just do a vector sum of B from different sources The figure shows four wires located at the 4 corners of a square. They carry equal currents in directions I-IN indicated What is the direction of B at the center of the square? I-OUT

I-IN B Coulombs Law For E-Fields When we computed the electric field due to charges we used Coulombs law. If one had a large irregular object, one broke it into infinitesimal pieces and computed dE, Which we write as, If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart. Jean-Baptiste Biot (1774-1862) The Biot-Savart Law For B-Fields

Quantitative rule for computing the magnetic field from any electric current Choose a differential element of wire of length dL and carrying a current i The field dB from this element at a point located by the vector r is given by the Biot-Savart Law 0 =4107 Tm/A (permeability constant of free space) i Felix Savart

(1791-1841) Biot-Savart Law for B-Fields Coulomb Law for E-Fields i Biot-Savart Requires A Right-Hand Rule Both Are 1/r2 Laws! The r has no units. Biot-Savart Law

An infinitely long straight wire carries a current i. Determine the magnetic field generated at a point located at a perpendicular distance R from the wire. Choose an element ds as shown Biot-Savart Law: dB points INTO the page Integrate over all such elements 0 ids( r sin q ) dB = 4 r3 0i ds(r sin q ) B=

3 4 - r Field of a Straight Wire 0 ids( r sin q ) dB = 3 4 r -q q - /2 sin q =cos(q - / 2) =R / r

r =( s2 + R2 )1/ 2 Rds 0i ds(r sin q ) 0i = B= 3 2 2 3/ 2 4 4 - r s +R ) - (

0i Rds = 2 2 3/ 2 2 s +R ) 0 ( 0iR s = 2 2

1/ 2 2 2 R (s + R ) 0 0i = 2R Is the B-Field From a Power Line Dangerous? A power line carries a current of 500 A. What is the magnetic field in a house located

100 m away from the power line? 0i B= 2R (4 10- 7 T m/ A)(500A) = 2 (100m) = 1 T Recall that the earths magnetic field is ~104T = 100 T Probably not dangerous! Biot-Savart Law A circular arc of wire of radius R carries a current i. What is the magnetic field at the

center of the loop? i Direction of B?? Not another right hand rule?! 0 idsR 0 iRdf dB = = 3 2 4 R 4 R 0 idf 0iF B= = 4 R 4 R

TWO right hand rules!: If your thumb points along the CURRENT, your fingers will point in the same direction as the FIELD. If you curl our fingers around direction of CURRENT, your thumb points along FIELD! Example, Magnetic field at the center of a circular arc of a circle pg 768: ICPP: What is the direction of the B field at point C? (a) Into the board ? (b) Out of the board ? (c) To the right ? (d) To the left ?

Example, Magnetic field at the center of a circular arc of a circle.: Example, Magnetic field off to the side of two long straight currents: