Source of Magnetic Field Ch. 28

Magnetic field of a moving charge B field of current element B field of current-carrying wire Force between conductors B field of circular current loop Ampere’s Law Applications of Ampere’s Law (sec. 28.1) (sec. 28.2) (sec. 28.3) (sec. 28.4) (sec. 28.5) (sec. 28.6) (sec. 28.7)

C 2009 J. Becker

(a) Magnetic field vectors caused by a moving positive point charge. At each point, B is perpendicular to the plane containing r and v. (b) Here the charge is moving into the screen.

Electric and magnetic forces on one of a pair of protons moving in E and B fields.

(a) Magnetic field vectors caused by current element dl. (b) In figure (b) the current is moving into the screen.

x

o

Magnetic field produced by a straight current carrying wire of length 2a. The direction of B at point P is into the screen.

Law of Biot and Savart

dB = [

m

o / 4

p

] [(I dL x r) / r

3

]

Magnetic field around a long, straight conductor. The field lines are circles, with directions determined by the right-hand rule.

Parallel conductors carrying currents in the same direction attract each other. The force on the upper conductor is exerted by the magnetic field caused by the current in the lower conductor.

Use Law of Biot and Savart, the integral is simple!

dB =

m

o

/ 4

p

(I dL x r) / r

3

Magnetic field caused by a circular loop of current. The current in the segment dL causes the field dB, which lies in the xy plane.

Ampere’s Law

Ampere’s Law states that the integral of B around any closed path equals m

o

times the current, I

encircled

, encircled by the closed loop.

We will use this law to obtain some useful results by choosing a simple path along which the magnitude of B is constant, (or independent of dl). That way, after taking the dot product, we can factor out |B| from under the integral sign and the integral will be very easy to do.

See list of important results in the Summary of Ch. 28 on p. 1094

Some (Ampere’s Law) integration paths for the line integral of B in the vicinity of a long straight conductor. Path in (c) is not useful because it does not encircle the current-carrying conductor.

To find the magnetic field at radius r < R, we apply Ampere’s law to the circle (path) enclosing the red area. For r > R, the circle (path) encloses the entire conductor.

x

o >

B =

m

o

n I, where n = N / L

A section of a long, tightly wound solenoid centered on the x-axis, showing the magnetic field lines in the interior of the solenoid and the current.

Coaxial cable. A solid conductor with radius a is insulated from a conducting rod with inner radius b and outer radius c.

Review

See

www.physics.edu/becker/physics51

C 2009J. Becker