1. Magnetism
Biot-Savart’s Law x R r q P I dx x • z R r dB q

2. Our Study of Magnetism Lorentz Force Eqn Motion in a uniform B-field
circular orbit
Forces on charges moving in wires
Magnetic dipole
Today: fundamentals of how currents generate magnetic fields
Next Time: Ampere’s Law  simplifies calculation

3. Today... Biot-Savart Law for Calculating B-field
Biot-Savart Law (“brute force”)
Ampere’s Law (“high symmetry”)
B-field of an ¥ Straight Wire
Force on Two Parallel Current
Carrying Wires
B-field of a Circular Loop

4. Last time:Potential Energy of Dipole
positive work
negative work B x m B x m B x m
t = 0
U = -mB
t = 0
U = mB
t = mB X
U = 0

5. Lecture 14, Act 1 (b) (a) (c) It will not rotate (a) U0 is minimumx y B a b
A circular loop of radius R carries current I as shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.
The coil will rotate to which of the following positions? 1A y y w w
(a)
(b)
(c) It will not rotate a b a b z z 1B
(a) U0 is minimum
(b) U0 is maximum
(c) neither
What is the potential energy U0 of the loop in its initial position?

6. Lecture 14, Act 1 (b) (a) (c) It will not rotatex y B a b
A circular loop of radius R carries current I as shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.
The coil will rotate to which of the following positions? 1A z y a b w z y a b w
(a)
(b)
(c) It will not rotate
The coil will rotate if the torque on it is non-zero:
The magnetic moment m is in +z direction.
Therefore the torque t is in the +y direction.
Therefore the loop will rotate as shown in (b).

7. Lecture 14, Act 1 (a) U0 is minimum (b) U0 is maximum (c) neithery B a b
A circular loop of radius R carries current I as shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.
What is the potential energy U0 of the loop in its initial position? 1B
(a) U0 is minimum
(b) U0 is maximum
(c) neither
The potential energy of the loop is given by:
In its initial position, the loop’s magnetic moment vector points in the +z direction, so initial potential energy is ZERO
This does NOT mean that the potential energy is a minimum!!!
When the loop is in the y-z plane and its magnetic moment points in the same direction as the field, its potential energy is NEGATIVE and is in fact the minimum.
Since U0 is not minimum, the coil will rotate, converting potential energy to kinetic energy!

8. Calculation of Electric Field
Two Ways to calculate
"Brute force"
Coulomb’s Law
"High symmetry" q S d E = ò r e
Gauss’ Law
What are the analogous equations for the
Magnetic Field?

9. Calculation of Magnetic Field
Two Ways to calculate
Biot-Savart Law
(“Brute force”)
Ampere’s Law
(“High symmetry”) I
AMPERIAN SURFACE/LOOP
These are the analogous equations

10. Biot-Savart Law... ...bits and piecesdl q r X dB
The magnetic field “circulates” around the wire
Use right-hand rule: thumb along I, fingers curl in direction of B.

11. +x -x +y -y +z -z Preflight 14:
A current carrying wire (with no remarkable symmetry) is oriented in the x-y plane. Points A,B, & C lie in the same plane as the wire. The z-axis points out of the screen.
5) In what direction is the magnetic field contribution from the segment dl at point A. Check all non-zero components.
+x x y y z z
6) In what direction is the magnetic field contribution from the segment dl at point B. Check all non-zero components.
7) In what direction is the magnetic field contribution from the segment dl at point C. Check all non-zero components.

12. dB points in the direction of
A: dl is to the right, and r is up  dB is out of the page
B: dl is to the right, and r is up and right  dB is out of the page
C: dl is to the right, and r is down and right  dB is into the page
Conclusion: at every point above the wire, dB is . Below the wire , dB is 

13. Preflight 14:
9) Would any of your answers for questions 5-7 change if we integrated dl over the whole wire?
NO!
Why or why not?
dB points in the direction of
At point A: dl is to the right, and r is up  dB is out of the page
At point B: dl is to the right, and r is up and right  dB is out of the page
At point C: dl is to the right, and r is down and right  dB is into the page
For every point in the x-y plane and every piece of wire dl:
every dl and every r are always in the x-y plane
Since dB must be perpendicular to r and dl, dB is always in the ±z direction!
Conclusion:
At every point above the wire, the dB due to every piece dl is .
Below the wire, the dB due to every piece dl is 

14. B-field of ¥ Straight Wirey P
Calculate field at point P using Biot-Savart Law: q r R q x
Which way is B?  (+z ) I dx
Its result is:
This calculation appears in its entirety in the appendix
Units of B are Tesla
1 Tesla = 104 Gauss
Earth’s B-field ~ 0.5 G

15. Putting it all together
We know that a current-carrying wire can experience force from a B-field.
We know that a a current-carrying wire produces a B-field.
Therefore: We expect one current-carrying wire to exert a force on another current-carrying wire:
Current goes together  wires come together
Current goes opposite  wires go opposite d Ia Ib

16. F on 2 Parallel Current- Carrying Wires• d Ib Ia
Calculate force on length L of wire b due to field of wire a:
The field at b due to a is given by:
Magnitude of F on b = Þ × d Ib Ia
Calculate force on length L of wire a due to field of wire b:
The field at a due to b is given by:
Magnitude of F on a = Þ

17. Preflight 14:
2) Two slack wires are carrying current in opposite directions. What will happen to the wires? They will:
a) attract
b) repel
c) twist due to torque
3) Now, two slack wires are carrying current in the same direction. What will happen to the wires? They will:
a) attract
b) repel
c) twist due to torque

18. a) Find B due to one wire at the position of the other wire
b) Use to find the direction of F in each case
a: Point your thumb down, your fingers wrap in the direction of B around the wire: The direction of B due to the left wire at the position of the right wire is out of the screen.
b: I is up, B is out of the screen, so F is to the right.  the force is repulsive.

19. Lecture 14, Act 2A (a) Fx < 0 (b) Fx = 0 (c) Fx > 0
Ftop
Fbottom X I
A current I flows in the +y direction in an infinite wire; a current I also flows in the loop as shown in the diagram.
What is Fx, the net force on the loop in the x-direction? y
Fright
Fleft I
(a) Fx < 0
(b) Fx = 0
(c) Fx > 0 x
You may have remembered a previous act in which the net force on a current loop in a uniform B-field is zero, but the B-field produced by an infinite wire is not uniform!
Forces cancel on the top and bottom of the loop.
Forces do not cancel on the left and right sides of the loop.
The left segment is in a larger magnetic field than the right
Therefore, Fleft > Fright

20. Lecture 14, Act 2B What is the magnitude of the magnetic
field at the center of a loop of radius R,
carrying current I?
(a) B = 0
(b) B = (m0I)/(2R)
(c) B = (m0I)•(2pR)
We can immediately guess the right answer must be (b).
(a) is wrong, because all parts of the loop contribute to a B going into the paper at the center of the loop.
(c) also cannot be correct—it has the wrong units:

21. Lecture 14, Act 2 But how do we calculate B?I
But how do we calculate B?
We must use Biot-Savart Law to calculate the magnetic field at the center: R
Two nice things about calculating B at the center of the loop:
I dl is always perpendicular to r: r into page
r is constant (r = R)
Note:

22. Circular Loop, in more detail
Circular loop of radius R carries current I. Calculate B along the axis of the loop: x • z R r z dB q r dB
Magnitude of dB from element dl:
dB = m I 4 p dl 2 = z +R
What is the direction of the field?
Symmetry Þ B in z-direction. B z = m I 4 p R (z 2 +R )
3/2 õ ó dl dB
cos q Þ

23. Circular Loop, cont. dl = 2 p R Þ B = m I 4 (z +R ) x • z R r dB q B =õ ó dl
= 2 p R Þ B z = m I 4 (z 2 +R )
3/2 x • z R r dB q B z = m IR 2
2(z +R )
3/2 B z m IR 2 2z 3
Note the form the field takes for z >> R:
Expressed in terms of the magnetic moment: m
= I p R 2 Þ B z 3
note the typical dipole field behavior!

24. Circular Loop, anywhere on axis
Expressed in terms of the magnetic moment R z B
Note the typical 1/z3 dipole field behavior!

25. Lecture 14, Act 3 (a) Bz(A) < 0 (b) Bz(A) = 0 (c) Bz(A) > 0x o z I A B
Equal currents I flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.
What is the magnetic field Bz(A) at point A, the midpoint between the two loops? 3A
(a) Bz(A) < 0
(b) Bz(A) = 0
(c) Bz(A) > 0 3B
What is the magnetic field Bz(B) at point B, just to the right of the right loop?
(a) Bz(B) < 0
(b) Bz(B) = 0
(c) Bz(B) > 0

26. Lecture 14, Act 3 (a) Bz(A) < 0 (b) Bz(A) = 0 (c) Bz(A) > 0x o z I A B
Equal currents I flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.
What is the magnetic field Bz(A) at point A, the midpoint between the two loops? 3A
(a) Bz(A) < 0
(b) Bz(A) = 0
(c) Bz(A) > 0
The right current loop gives rise to Bz <0 at point A.
The left current loop gives rise to Bz >0 at point A.
From symmetry, the magnitudes of the fields must be equal.
Therefore, B(A) = 0

27. Lecture 14, Act 3 (a) Bz(B) < 0 (b) Bz(B) = 0 (c) Bz(B) > 0
Equal currents I flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.
What is the magnetic field Bz(B) at point B, just to the right of the right loop? 3B
(a) Bz(B) < 0
(b) Bz(B) = 0
(c) Bz(B) > 0
The signs of the fields from each loop are the same at B
as they are at A!
However, point B is closer to the right loop, so its field wins!

28. Summary Biot-Savart Law for Calculating B-Fields
Current-Carrying Wires Make B-Fields and Are Affected by B-Fields
Current Loop Produces a Dipole Field

29. Appendix A: B-field of ¥ Straight Wirey P
Calculate field at point P using Biot-Savart Law: q r R q x
Which way is B? I +z dx
Rewrite in terms of R,q : Þ Þ
therefore, Þ

30. Appendix A: B-field of ¥ Straight Wireq P I dx Þ
therefore,