Sources of Magnetic Fields

1. Sources of Magnetic Fields AP Physics C Chapter 28

2. Magnetic Field of a Moving Charge

3. Magnetic Field of a Moving Charge • http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html • Experiments support the following observations about the magnetic field: • Proportional to the quantity of charge • Proportional to inverse square of the distance from the source point to the field point • Proportional to the sine of the angle between the velocity vector and the position vector • Proportional to the speed of the moving charge

4. Magnetic Field of a Moving Charge • Further observations about the magnetic field: • B-Field is perpendicular to the plane containing the line from the source point to the field point and the particle’s velocity • E-Field lines radiate outwards • B-Field lines are circles with centers along the line of v and lying in the planes perpendicular to this line • RHR is used to determine the direction of the B-Field • μo is called the permeability of free space and is equal to 4π x 10-7 T-m/A

5. Magnetic Field of a Current Element

6. Magnetic Field of a Current Element • If the cross-sectional area of the current element is A, the volume of the segment is Adl. If there are n charges of q per unit volume, then the total charge of moving charges is dQ = nqAdl. The charges are moving with a drift velocity of vd. • This is known as Biot-Savart Law.

7. Properties of the Magnetic Field Created by an Electric Current: • The vector dB is perpendicular both to dl (which points in the direction of the current) and to the unit vector • The magnitude of dB is inversely proportional to r2 • The magnitude of dB is proportional to the current and to the magnitude dl of the length element dl • The magnitude of dB is proportional to sin θ, where θ is the angle between the vectors dl and • The Biot-Savart law is also valid for current consisting of charges flowing through space

8. Sample Problem #1 • Magnetic Field of a Straight Conductor • Given a straight conductor with length 2a carrying a current of I, find the B-field at a point on its perpendicular bisector, at a distance of x from the conductor.

9. Sample Problem #2 • Calculate the magnitude of the magnetic field 4.0 cm from an infinitely long, straight wire carrying a current of 5.0 A.

10. Sample Problem #3 • The diagram below is an end view of two long, straight, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite directions. Find the magnitude and direction of B at point P. P ddd

11. Force Between Parallel Conductors • Case 1: Current in the same direction • B-field due to current of conductor 1 • Force acting on conductor 2

12. Definition of the Ampere: • One Amp is that current which, when flowing in each of two straight, parallel, infinitely long wires, separated by 1m, in a vacuum, produces a force per unit of 2 ×10-7Nm-1.

13. Force Between Parallel Conductors • What is the force if the current is in the opposite directions? • If the current in one wire is 2 A and the other current is 6 A, which is true: • The force on the second wire is 3 times the force on the first wire. • The force on the first wire is 3 times the force on the second wire. • The forces are equal and opposite to each other. • A loose spiral spring is hung from the ceiling, and a large current is sent through it. Do the coils move closer together or farther apart?

14. Sample Problem #4 • The current in the long, straight wire is 5.00 A, and the wire lies in the plane of a rectangular loop, which carries 10.0 A. The long wire is 0.100 m from the rectangular loop, and the rectangular loop is 0.150 m x 0.450 m. Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire.

15. Magnetic Field of a Circular Loop • Given a single loop of wire, with a radius a, use the Biot-Savart law to find the magnetic field at point P.

16. Sample Problem #4 • A wire carrying a current of 5.00 A is to be formed into a circular loop of one turn. If the required value of the magnetic field at the center is 10.0 μT, what is the required radius?

17. Sample Problem #5 • A coil consisting of 100 circular loops with radius 0.60 m carries a current of 5.0 A. • Find the magnetic field at a point along the axis of the coil, 0.80 m from the center. • Along the axis, at what distance from the center is the field magnitude 1/8 as great as it is at the center?

18. Ampere’s Law • Provides an alternative formulation of the relation between a magnetic field and its source. • Used to find magnetic fields caused by some highly symmetric current distributions and to find current distributions corresponding to particular magnetic field configurations. • Analogous to Gauss’s law for determining electric fields of highly symmetric distributions of charge. • Ampere’s law valid for steady current and magnetic fields that do not vary with time.

19. Ampere’s Law • Consider a magnetic field caused by a long, straight conductor carrying a current I: • Ampere’s Law:

20. Sample Problem #6 • A long, straight wire of radius R carries a steady current I that is uniformly distributed through the cross-section of the wire. Calculate the magnetic field a distance r from the center of the wire in the regions • r > R • r < R • Sketch a B vs. r graph

21. Sample Problem #7 • A solenoid is a long wire wound in the form of a helix. An ideal solenoid is approached when the turns are closely spaced and the length is much greater than the radius of the turns. In this case, the external field is zero, and the interior field is uniform over a great volume. Find the magnetic field inside a solenoid.

22. Toroidal Solenoid • Consider a toroidal solenoid having N closely spaced turns of wire, calculate the magnetic field inside the solenoid and outside of the solenoid.