Torque on a Current Loop, 2

1. Torque on a Current Loop, 2 • There is a force on sides 2 & 4 since they are perpendicular to the field • The magnitude of the magnetic force on these sides will be: • F2 = F4 = I a B • The direction of F2 is out of the page • The direction of F4 is into the page

2. Torque on a Current Loop, 3 • The forces are equal and in opposite directions, but not along the same line of action • The forces produce a torque around point O

3. Torque on a Current Loop, Equation • The maximum torque is found by: • The area enclosed by the loop is ab, so τmax = IAB • This maximum value occurs only when the field is parallel to the plane of the loop

4. Torque on a Current Loop, General • Assume the magnetic field makes an angle of q< 90o with a line perpendicular to the plane of the loop • The net torque about point O will be τ = IAB sin q • Use the active figure to vary the initial settings and observe the resulting motion PLAY ACTIVE FIGURE

5. Torque on a Current Loop, Summary • The torque has a maximum value when the field is perpendicular to the normal to the plane of the loop • The torque is zero when the field is parallel to the normal to the plane of the loop • where is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop

6. Direction • The right-hand rule can be used to determine the direction of • Curl your fingers in the direction of the current in the loop • Your thumb points in the direction of

7. Magnetic Dipole Moment • The product Iis defined as the magnetic dipole moment, , of the loop • Often called the magnetic moment • SI units: A · m2 • Torque in terms of magnetic moment: • Analogous to for electric dipole

8. Chapter 30 Sources of the Magnetic Field

9. Biot-Savart Law – Introduction • Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet • They arrived at a mathematical expression that gives the magnetic field at some point in space due to a current

10. Biot-Savart Law – Set-Up • The magnetic field is at some point P • The length element is • The wire is carrying a steady current of I Please replace with fig. 30.1

11. Biot-Savart Law – Observations • The vector is perpendicular to both and to the unit vector directed from toward P • The magnitude of is inversely proportional to r2, where r is the distance from to P

12. What does this tell you about the magnetic field, ? • It goes like the scalar dot product of and • It goes like X • is usually zero 0 of 30

13. Biot-Savart Law – Observations, cont • The magnitude of is proportional to the current and to the magnitude ds of the length element • The magnitude of is proportional to sin q, where q is the angle between the vectors and

14. Biot-Savart Law – Equation • The observations are summarized in the mathematical equation called the Biot-Savart law: • The magnetic field described by the law is the field due to the current-carrying conductor • Don’t confuse this field with a field external to the conductor

15. Permeability of Free Space • The constant mo is called the permeability of free space • mo = 4p x 10-7 T. m / A

16. Total Magnetic Field • is the field created by the current in the length segment ds • To find the total field, sum up the contributions from all the current elements I • The integral is over the entire current distribution

17. Biot-Savart Law – Final Notes • The law is also valid for a current consisting of charges flowing through space • represents the length of a small segment of space in which the charges flow • For example, this could apply to the electron beam in a TV set

18. Compared to • Distance • The magnitude of the magnetic field varies as the inverse square of the distance from the source • The electric field due to a point charge also varies as the inverse square of the distance from the charge

19. Compared to , 2 • Direction • The electric field created by a point charge is radial in direction • The magnetic field created by a current element is perpendicular to both the length element and the unit vector

20. Compared to , 3 • Source • An electric field is established by an isolated electric charge • The current element that produces a magnetic field must be part of an extended current distribution • Therefore you must integrate over the entire current distribution

21. Which variable can be pulled out of the integral? • ds • sinθ • r2 • None of them 0 of 30

22. How are θ and Φ related? • Φ = θ – π/2 • Φ = θ • Φ = π/2 – θ • Φ = θ + π/2 0 of 30

23. for a Long, Straight Conductor, Special Case • The field becomes

24. for a Long, Straight Conductor, Direction • The magnetic field lines are circles concentric with the wire • The field lines lie in planes perpendicular to to wire • The magnitude of the field is constant on any circle of radius a • The right-hand rule for determining the direction of the field is shown

25. for a Curved Wire Segment • Find the field at point O due to the wire segment • I and R are constants • q will be in radians

26. What about the contribution from the wires coming in and going out? • They are distant enough to neglect their contribution • X = 0 • The two currents cancel each other 0 of 30

27. for a Curved Wire Segment • Find the field at point O due to the wire segment • I and R are constants • q will be in radians

28. for a Circular Loop of Wire • Consider the previous result, with a full circle • q = 2p • This is the field at the center of the loop

29. for a Circular Current Loop • The loop has a radius of R and carries a steady current of I • Find the field at point P

30. What can we pull out of the integral this time? • r2 • Sin θ • ds • nothing 0 of 30

31. for a Circular Current Loop • The loop has a radius of R and carries a steady current of I • Find the field at point P

32. Comparison of Loops • Consider the field at the center of the current loop • At this special point, x = 0 • Then, • This is exactly the same result as from the curved wire