1. Physics 202, Lecture 13 Today’s Topics Magnetic Forces: Hall Effect (Ch. 27.8) Sources of the Magnetic Field (Ch. 28) B field of infinite wire Force between parallel wires Biot-Savart Law Examples: ring, straight wire Ampere’s Law: infinite wire, solenoid, toroid

2. The Hall Effect (1) positive charges moving counterclockwise: upward force, upper plate at higher potential negative charges moving clockwise: upward force Upper plate at lower potential Equilibrium between electrostatic & magnetic forces: Potential difference on current-carrying conductor in B field:

3. The Hall Effect (2) Hall coefficient: Hall effect: determine sign,density of charge carriers (first evidence that electrons are charge carriers in most metals) Text: 27.47

4. Magnetic Fields of Current Distributions Two ways to calculate E directly: "Brute force" – Coulomb’s Law "High symmetry" – Gauss’ Law  0 = 8.85x10 -12 C 2 /(N m 2 ): permittivity of free space Recall electrostatics: Coulomb’s Law) (point charges)

5. Magnetic Fields of Current Distributions  I – Biot-Savart Law (“Brute force”) –AMPERIAN LOOP 2 ways to calculate B: – Ampere’s Law (“high symmetry”)  0 = 4  x10 -7 T  m/A: permeability of free space (2 circuits)

6. Magnetic Field of Infinite Wire closed circular loops centered on current Result (we’ll derive this both ways): right-hand rule

7. Which figure represents the B field generated by the current I ? x x x....... x x x... x x x... Quick Question 1: Magnetic Field and Current

8. Quick Question 2: Magnetic Field and Current Which figure represents the B field generated by the current I ?

9. Forces between Current-Carrying Wires A current-carrying wire experiences a force in an external B -field, but also produces a B -field of its own. d IaIa IbIb d IaIa IbIb Parallel currents: attract Opposite currents: repel Magnetic Force:

10. Example: Two Parallel Currents Force on length L of wire b due to field of wire a : d IaIa IbIb x Force on length L of wire a due to field of wire b : Text: 28.4, 28.58

11. Biot-Savart Law... I B field “circulates” around the wire Use right-hand rule: thumb along I, fingers curl in direction of B. X dB dl r  …add up the pieces

12. B Field of Straight Wire, length L (Text example 28-13) B field at point P is:  When L=infinity (text example 28-11):   Text: 28.43 (return to this later with Ampere’s Law)

13. B Field of Circular Current Loop on Axis Use Biot-Savart Law (text example 28-12) See also: center of arc (text example 28-13) Text: 28.37

14. B of Circular Current Loop: Field Lines B

15. Ampere’s Law Ampere’s Law:  applies to any closed path, any static B field  useful for practical purposes only for situations with high symmetry  Generalized form (valid for time-dependent fields): Ampere-Maxwell (later in course) I dldl

16. Use symmetry and Ampere’s Law: Ampere’s Law: B-field of  Straight Wire  I Quicker and easier method for B field of infinite wire (due to symmetry -- compare Gauss’ Law) Choose loop to be circle of radius R centered on the wire in a plane  to wire. Why? Magnitude of B is constant (function of R only) Direction of B is parallel to the path.

17. B Field Inside a Long Wire Total current I flows through wire of radius a into the screen as shown. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x a r  What is the B field inside the wire? By symmetry -- take the path to be a circle of radius r : Current passing through circle: Text example: 28-6

18. B Field of a Long Wire Outside the wire: ( r > a ) Inside the wire: ( r < a ) r B a Text: 28.31

19. Ampere’s Law: Toroid (Text example 28-10) N = # of turns Show:

20. Ampere’s Law: Toroid Toroid: N turns with current I. B  =0 outside toroid! B  inside: consider circle of radius r, centered at the center of the toroid. x x x x x x x x x x x x x x x x r B (Consider integrating B on circle outside toroid: net current zero)

21. B Field of a Solenoid Inside a solenoid: source of uniform B field If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. Solenoid: current I flows through a wire wrapped n turns per unit length on a cylinder of radius a and length L. L a To calculate the B-field, use Biot-Savart and add up the field from the different loops. In this limit, can calculate the field using Ampere's Law!

22. Ampere’s Law: Solenoid The B field inside an ideal solenoid is: ideal solenoid n=N/L segment 3 at 

23. Right-hand rule: direction of field lines Solenoid: Field Lines Like a bar magnet! (except it can be turned on and off) Text: 28.68 north polesouth pole