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. Text Reference: Chapter 29.1 through 29.3
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
Text Reference: Chapter 29.1 through 29.3

4. 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?

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

6. 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.

7. 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

8. 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

9. 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

10. Lecture 14, Act 1 (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

11. Lecture 14, Act 2 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:

12. 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:

13. 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

14. 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!

15. Circular Loop R B z 1 z 3 z B z = m iR 2
2(z +R )
3/2

16. 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

17. 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

18. 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!

19. Text Reference: Chapter 29.1 through 29.4
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
Text Reference: Chapter 29.1 through 29.4

20. Appendix: 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

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