Chapter 28

1. Chapter 28 Sources of Magnetic Field

2. Goals for Chapter 28 • To study the magnetic field generated by a moving charge • To consider magnetic field of a current-carrying conductor • To examine the magnetic field of a long, straight, current-carrying conductor • To study the magnetic force between current-carrying conductors • To consider the magnetic field of a current loop • To examine and use Ampere’s Law

3. Chapter HW • Read the chapter and do the HW: • 11, 19, 21, 23, 24, 35, 37, 39

4. Introduction • Normally, when someone describes a solenoid, they are likely to use a doorbell or car-starter as their example. In the photo at right, scientists at CERN are using the most powerful magnetic field ever proposed.

5. magnetic field of moving charge • Like an electric field, the magnetic field of a moving charge is proportional to the magnitude of the charge and the field points distance from the source point.

6. Vector B and definition of μo • The magnetic field of a point charge with constant velocity can be found by: • An accelerating charge introduces many complications that are beyond the scope of this course. • The proportionality constant: • From section 21.3 we said that the constant 1/4πεo was equal to (10-7 N*s2/C2)c2. This can be rearranged to c2=1/μoεo. • This gives a value for the speed of light in a vacuum.

7. Moving charges—field lines • The moving charge will generate field lines in circles around the charge in planes perpendicular to the line of motion. • Follow Example 28.1. • Refer to Figure 28.2.

8. 28.1 Summary • The magnetic field B created by a charge q moving with velocity v depends on the distance r from the source point (the location of q) to the field point (where B is measured). The B field is perpendicular to v and to r the unit vector directed from the source point to the field point. The principle of superposition of magnetic fields states that the total B field produced by several moving charges is the vector sum of the fields produced by the individual charges.

9. Magnetic field of a current element By the superposition of magnetic fields we can create an expression for the magnetic field of a current carrying wire. The derivation is on page 1068.

10. The Law of “Bee-oh” and “Suh-var” • The following equations are called the law of Biot and Savart. We can use these equations to find the magnetic field at any point in space due to the current in a complete circuit. We will follow Problem Solving Strategy on page 1069 for many examples.

11. Magnetic field of a current element II • Follow Example 28.2 and Figure 28.4 below.

12. 28.2 Summary and Homework • The law of Biot and Savart gives the magnetic field dB created by an element dl of a conductor carrying current I. The field dB is perpendicular to both dl and r the unit vector from the element to the field point. The B field created by a finite current-carrying conductor is the integral of dB over the length of the conductor. • Page 1096: 5&9

13. Magnetic field of a straight current-carrying conductor • Biot and Savart contributed to finding the magnetic field produced by a single current-carrying conductor.

14. Magnetic field of a straight current-carrying conductor • Let dl = dy. • The cross product is: Idy sin φ, but sin φ = sin(π-φ) because sin 80o = sin 100o. • We can substitute x/√(x2 + y2) for sin(π-φ), so our equation:

15. A long, straight, current-carrying conductor • When the length 2a of the conductor is very great in comparison to its distance x from point P, we can consider it to be infinitely long. In this case, the a’s cancel, leaving B = μoI/2πx. • We can generalize this for a situation where we can calculate the magnetic field for any field point a distance r from the wire:

16. Fields around single wires • Refer to Example 28.3. • Refer to Example 28.4. • Figure 28.7 illustrates Example 28.4. • These apply to wires like the one at right in Figure 28.8.

17. 28.3 Summary and Homework • The magnetic field B at a distance r from a long, straight conductor carrying a current I has a magnitude that is inversely proportional to r. The magnetic field lines are circles coaxial with the wire, with directions given by the right-hand rule. • Read 1074 to 1086 • Page 1097: 11, 17, 19

18. Forces and parallel conductors • In the figure to the right we have two current carrying conductors with currents I and I’ where each lies in the magnetic field produced by the other. Each experiences a force from the other. • The lower produces a field felt by the upper with a magnitude of μoI/2πr. The force on a length L of the upper conductor: F = I’LB = μoIIL’/2πr. • The force per unit length is:

19. Forces and parallel conductors • The current in the upper conductor also sets up a field and hence a force per unit length on the lower wire. From the RHR the forces pull the wires together. • Parallel conductors with current in the same direction attract, while opposing current carrying conductors repel. • One ampere is that unvarying current that, if present in each of two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2 x 10-7 newtons per meter of length. • Try example 28.5.

20. 28.4 Summary and Homework • Two long, parallel, current-carrying conductors attract if the currents are in the same direction and repel if the currents are in opposite directions. The magnetic force per unit length between the conductors depends on their currents I and I’ their separation r. The definition of the ampere is based on this relationship. • Page 1098: 21, 23

21. Magnetic field of a circular current loop • The current I enters the loop of radius a and leaves the loop through straight conductors with opposite direction of current. The magnetic field from these wires cancel out. • At point P, all magnetic field components that do not lie on the x-axis cancel out. The distance r = √(x2 + a2). At all points, dl and r are perpendicular, so φ = 90, and sin φ = 1. So dBx = dB cos θ • Integrate to find Bx. • For a coil with N number of loops, the magnetic field increases by a factor of N. • At the center of N number of loops:

22. Magnetic fields in coils • We can express the magnetic field in terms of the magnetic dipole moment μ = IA where A = πr2: • This shows that the magnetic field is directly proportional to the magnetic dipole moment, and that the field is from the dipole moment. The direction of the field is the same as the direction of the dipole moment. • Try example 28.7.

23. 28.5 Summary and Homework • The law of Biot and Savart allows us to calculate the magnetic field produced along the axis of a circular conducting loop of radius a carrying current I. The field depends on the distance x along the axis from the center of the loop to the field point. If there are N loops, the field is multiplied by N. At the center of the loop, x = 0. • Page 1098: 25 & 29

24. Ampere’s Law I—specific then general • Ampere’s law states that the closed surface integral of the magnetic field B around a current carrying wire doted into a length vector dl is equal to μo times the current . • When there is more than one current carrying wire, we need to update this equation by stating that the current is the total current enclosed in the closed surface. • We also need to take into account the direction of the current. We have another RHR to help us out: Curl your fingers in the direction of dl, you thumb points in the direction of a positive current.

25. CAUTION • In Chapter 23 we saw that the line integral of the electrostatic field E around any closed path is equal to zero; this is a statement that the electrostatic force F = qE on a point charge q is conservative, so this force does zero work on a charge that moves around a closed path that returns to the starting point. You might think that the value of the line integral is similarly related to the question of whether the magnetic force is conservative. This isn’t the case at all. Remember that the magnetic force on a moving charged particle is always perpendicular to B, so the line integral is not related to the work done by the magnetic force; as stated in Ampere’s law, this integral is related only to the total current through a surface bounded by the integration path. In face, the magnetic force on a moving charged particle is not conservative. A conservative force depends only on the position of the body on which the force is exerted, but the magnetic force on a moving charged particle also depends on the velocity of the particle.

26. Ampere’s Law II • Consider the figure. • Follow Problem-Solving Strategy on page 1083. • Follow Example 28.8.

27. Field inside a long cylindrical conductor • A cylinder of radius R carrying a current I. • Refer to Example 28.9 and Figure 28.18 and Figure 28.19.

28. 28.10 Field of a solenoid

29. 28.11 Field of a toroidal solenoid

30. 28.6 & 7 Summary and Homework • Ampere’s law states that the line integral of B around any closed path equals μo times the net current through the area enclosed by the path. The positive sense of current is determined by a right-hand rule. • Page1098: 31, 35, 37, 39