1. Magnetic Fields due to Currents
Physics 014
Magnetic Fields due to Currents

2. Topics Calculating the field Force between two currents Ampere’s Law
Solenoids and Toroids
Coil as a dipole

3. Calculating a wire’s field
Previously, we calculated the differential electric field due to a differential element of charge

4. Calculating a wire’s field

5. Calculating a wire’s field
Likewise, we can calculate the differential magnetic field due to a differential element of current

6. Calculating a wire’s field

7. Calculating a wire’s field
The magnitude of the field produced at point P by a current-length element is

8. Calculating a wire’s field
μ0 is the permeability constant
μ0 =4π x 10-7 T m/A
= 1.26x10-6 T m/A

9. Calculating a wire’s field
In vector form, we have the Biot-Savart law

10. Calculating a wire’s field
We can use Biot-Savart to calculate the magnitude of the magnetic field a distance R from a long (infinite) straight wire carrying current i

11. Calculating a wire’s field

12. Calculating a wire’s field
Use the right hand rule to get the direction of the field lines due to the current

13. Calculating a wire’s field

14. Calculating a wire’s field

15. Long wire Calc.

16. Calculating a wire’s field
Find the field produced by a circular arc of wire. The arc is through an angle φ, has radius R center C and carries current i.

17. Calculating a wire’s field
At C, all elements produce field dB
Angle between ds and r always 90 degrees
r=R always

18. Wire’s field (arc)

20. ???
The figure shows 3 circuits of circular arcs (radii r,2r,3r). The circuits carry the same current. Rank them according to the magnitude of the field lines at the center of curvature, greatest first.

21. ???
For three long, straight, parallel equally spaced wires with identical currents, Rank the wires according to the magnitude of the force on each due to the currents in the other two

22. Ampere’s Law
For systems with symmetry, we may use Ampere’s Law

23. Ampere’s Law Current enclosed Magnetic field
Differential element of path length

24. Ampere’s Law

25. Ampere’s Law

26. Ampere’s Law
Curl your right hand in the direction of ds to get the positive direction of the currents

27. Ampere’s Law How is this equation used?
Typically to figure out B for a given current
Short cut (use it, not Biot Savart)

28. Ampere’s Law
i enclosed

29. Ampere’s Law
Let’s now use Ampere’s law to calculate the magnetic field B at a point a distance r from a wire carrying current i.

30. Ampere’s Law
(Think gaussian surface)

31. Ampere’s Law Problem has cylindrical symmetry
Exploit symmetry with Ampere’s law
Do this by saying, B is the same at all points r away from wire and create Amperian loop

32. Ampere’s Law
B and ds same direction

33. ???
For three equal currents i and four Amperian loops, rank the loops (greatest first) according to the magnitude of

34. Solenoids and Toroids
A solenoid is simply a straight coil of wire with many loops

35. Solenoids and Toroids
Look at the magnetic field for a stretched out solenoid

36. Solenoids and Toroids
For a real solenoid

37. Solenoids and Toroids
We can use Ampere’s law to calculate the magnetic field within a solenoid

38. Solenoids and Toroids
Amperian loop

39. Solenoids and Toroids
From the Amperian loop (rectangle abcda)

40. Solenoids and Toroids The current enclosed is then
Ampere’s law then gives
n = number of turns in Amperian
loop

41. Solenoids and Toroids
A Toroid is a solenoid bent into a circle

42. Solenoids and Toroids
We can use Ampere’s law to calculate the magnetic field within a toroid

43. Solenoids and Toroids

44. Solenoids and Toroids
For a point r from the center of a toroid with N loops carrying a current I,

45. Coil as a Magnetic Dipole
Recall that we considered a coil of current carrying wire a dipole
Using Biot Savart we can calculate the magnetic field of the coil along the z axis

46. Coil as a Magnetic Dipole

47. Coil as a Magnetic Dipole

48. Coil as a Magnetic Dipole
After a bit of calculation,