College Physics 112: Sections 010, 011 – Spring 2006 Lecture 20: Sources of Magnetism – Ampere’s Law: We now know that currents produce magnetic fields. But what do these fields look like? Again in analogy to electric fields we draw lines of force to illustrate magnetic fields. We give a second name to the magnetic field B . We also call it the magnetic flux density and we calculate magnetic flux in a similar fashion to electric flux. That is: Def. Magnetic Flux: The magnetic flux is the number of magnetic field lines passing through a given area. For B constant: mag B A ; for non-constant fields: mag B d A . A key different between magnetic lines of force & electric lines of force is that magnetic lines of force have no beginning or end. Since there are no magnetic monopoles, magnetic lines of force are closed loops. Consider a long-straight wire carrying a current I. From the symmetry of the wire the magnetic field strength can only be a function of ‘r’ the radial distance from the wire. Hence, the strength of B at a distance ‘r’ in any direction must be the same. Thus the ‘lines of force’ are concentric circles with the wire at the center. We choose the direction I of B by a ‘Right-Hand-Rule’. Place your right hand on I the wire with thumb in the direction of the current. The B lines circle in the direction of your fingers. This is B B shown in the figures. The second figure has the current ‘I’ coming ‘out of the page.” Magnetic Field of a Long Straight Wire: B lsw
o I
2 r
Forces between wires: Consider two long straight parallel wires carrying currents I1 and I2. Let the distance between the wires be ‘r’. Since a current carrying wire interacts with a magnetic field, then we will have forces on the two wires. Wire #1 is sitting in the filed of wire #2, and vice-versa. In the figure the two wires carry the indicated currents into the paper. The magnetic lines of force of wire #1 are shown. Hence the direction of the field due to wire #1 at the location of #2 is to the left. The force exerted on #2 is given by: F wire I l B where I = I2 and the I magnitude of B is μ0 I1/2π r . By the right hand rule, the direction of F1 on 2 is upward. Of course we cannot solve for the force itself since we are considering the wires as ‘long’. Hence we solve for the force
I I per unit length. The result is: F1 on 2 / l o 1 2 . 2 r
1
B
I2
Def: The Ampere: One ampere is that steady current which when maintained in two parallel long straight wires separated a distance of 1 meter results in a force per unit length of exactly 2 x 10-7N/m Def: The Coulomb: One coulomb is the amount of charge that passes a cross-section of a wire carrying a current of 1 ampere in 1 second.
Ampere’s Law is analogous to Gauss’s Law in that it permits determination of Magnetic Fields for highly symmetric current distributions. Ampere’s Law: (mathematical form)
B d l o I net where the sum is around a closed path. Inet
path
is the net current passing through the area defined by the path. The path is always selected in a clockwise direction making a current Iinto a positive current and an Iout A negative current. The application of symmetry principles tells you the correct path to use such that the field strength is constant along the path. E. g. for a long straight wire we know B= B(r) i.e., can only depend on ‘r’. A path of constant ‘r’ is a circle. Thus choosing a circle of radius ‘r’ about our wire (see figure above) we have on the left hand side: B d l B dl cos 0 . Hence the ‘B’ comes outside & the left hand side is simply B 2π r . The net current (into) is simply I . Thus solving for B we obtain the answer for the field due to a long straight wire.
Key Concepts: Memorize the definition of magnetic flux. Memorize the result for B due to a long straight wire. Memorize the definition of the ampere. Memorize the mathematical statement of Ampere’s Law. What is Ampere’s Law useful for?