1. Today: fundamentals of how currents generate magnetic fields
Our Study of Magnetism
Lorentz Force Equation
Motion in a uniform B-field
Forces on charges moving in wires
Magnetic dipole N S
Today: fundamentals of how currents generate magnetic fields
11/22/2018

2. LECTURE 14 Fundamental Laws for Calculating B-field
Biot-Savart Law (“brute force”)
Ampere’s Law (“high symmetry”)
Example: B-field of an Infinite Straight Wire
from Biot-Savart Law
from Ampere’s Law
Other examples

3. Biot-Savart Law: B-field due to a current in a wire
Biot-Savart Law: B-field due to a moving charge
Biot-Savart Law: B-field due to a current in a wire
Right Hand Rules
The magnetic field “curls” or “loops” around the wire.
No dB field at P2 since d is parallel to r.

4. Observations about Biot-Savart Law
11/22/2018

5. Oersted’s Experiment Switch closed: I 0 Switch open: I = 0
DEMO
6B-01
Oersted’s Experiment
Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand. While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon. But he couldn't! The needle was neither attracted to the wire nor repelled from it. Instead, it tended to stand at right angles (see drawing below). In the end he published his findings (in Latin!) without any explanation
Switch open: I = 0
Compass points north.
Switch closed: I 0
11/22/2018

6. Magnetic Field of an Infinite Straight Wire
Calculate field at point P using Biot-Savart Law: x dl φ I R P y dl r φ

7. Calculation of Electric Field
Two ways to calculate
Coulomb’s Law
"Brute force"
Gauss’ Law
"High symmetry"
What are the analogous equations for the
Magnetic Field?

8. Calculation of Magnetic Field
Two Ways to calculate
Biot-Savart Law I
"Brute force"
Ampere’s Law AMPERIAN LOOP INTEGRAL
"High symmetry“ also,
only the ENCLOSED Current
These are the analogous equations

9. Magnetic Field of an Infinite Straight Linedl R I B
Calculate field at distance R from wire using Ampere's Law:
Ampere's Law simplifies the calculation thanks to symmetry around the current! (axial/cylindrical)

10. Magnetic Field Lines of a Straight Wire
DEMO – 6B-03
11/22/2018

11. Example x 2I I a b y
A current I flows in an infinite straight wire in the +z direction (towards you). A concentric infinite cylinder of radius R carries current 2I in the -z direction.
What is the magnetic field Bx(a) at point a, just outside the cylinder as shown?
(a) Bx(a) < 0
(b) Bx(a) = 0
(c) Bx(a) > 0

12. Example x 2I I a b y
A current I flows in an infinite straight wire in the +z direction (towards you). A concentric infinite cylinder of radius R carries current 2I in the -z direction.
What is the magnetic field Bx(b) at point b, just inside the cylinder as shown?
(a) Bx(b) < 0
(b) Bx(b) = 0
(c) Bx(b) > 0

13. QUIZ lecture 14
An infinitely long hollow conducting tube carries current I in the direction shown. What is the direction of the magnetic field inside the tube?
The magnetic field is zero
Radially outward to the center
Radially inward from the center
Counterclockwise
Clockwise
11/22/2018

14. QUIZ lecture 14 R I
CASE 2 R
CASE 1 I
Two loops are placed near current carrying wires as shown in Case 1 and Case 2. In both cases, the direction of the current in the two wires are opposite to each other. For which loop is greater?
same
Case 1
Case 2
The integral is the same for both cases.
11/22/2018

15. Magnetic Field at the Center of a Current Loop
Practice both right hand rules here:
11/22/2018

16. Line Segment Combinations2 I
Find B at point P. 1 3 R I I P
11/22/2018

17. Only the arcs contribute.
Double Arc
Find B at point P.
Only the arcs contribute.
11/22/2018

18. Magnetic Field on Axis of Current Loop
11/22/2018

19. Magnetic Field Lines of a Current Loop
DEMO – 6B-04 & 05
11/22/2018

20. Solenoid (DEMO)
DEMO – 6B-04 & 05
11/22/2018

21. Magnetic Field of a Solenoid
Step 1: Cut up the distribution
into pieces
Step 2: Contribution of one piece
origin: center of the solenoid
one loop: B
Number of loops per meter: N/L
Number of loops in z: (N/L) z
Field due to z:

22. Magnetic Field of a Solenoid
Step 3: Add up the contribution
of all the pieces B
Magnetic field of a solenoid:

23. Magnetic Field of a Solenoid
Special case: R<<L, center of the solenoid (d~0):
in the middle of a long solenoid

24. Demos
6B-13

25. Earth’s Magnetic Field
11/22/2018

26. Earth’s Magnetic Field (continued)
How much current must the loop carry to produce a
magnetic field of 0.7 gauss at one of the Earth’s poles?
10kG = 1T
11/22/2018

27. Earth’s Magnetic Field (continued)
It has decreased by ~ 6% in the last century.
Last polarity reversal was ~ 750,000 years ago.
Time between reversals has varied between 20K and 37M years. Average is 500K years.
Inner temperature is between 3,900 and 7,2000 C degrees . It has cooled about 1100 C in the last 4 billion years.
There is some agreement that convective motion of the molten part of the earth is generating the magnetic field.
11/22/2018