1. Magnetic Fields due to Currents
Chapter 29
Magnetic Fields due to Currents

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

3. Magnetic Fields 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. 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 SURFACE/LOOP ∫ I
These are the analogous equations

6. The current element Idl produces the field dB @ the distance r:
Biot-Savart Law...
The current element Idl produces the field the distance r: I dl
Magnetic permeability of vacuum: 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. Biot Savart
Note:

8. +x -x +y -y +z -z Exercise Set 5-7:
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.

9. 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 

10. Exercise:
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 

11. Circular Loop Center 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. Center Calculation But how do we calculate B?
We must use Biot-Savart Law to calculate the magnetic field at the center: R
I dl is always perpendicular to r: out of page
r is constant (r = R)
Note:

14. Circular Loop, On Axis Off Center (optional)
Circular loop of radius R carries current I. Calculate B along the axis of the loop: x • z R r x dB  θ r dB
Magnitude of dB from element dl:
dB = m I 4 p dl 2 = x +R
What is the direction of the field?
Symmetry Þ B in x-direction. B x = m I 4 p R (x 2 +R )
3/2 õ ó dl dB
sin q Þ

15. Circular Loop, cont. (optional)õ ó dl
= 2 p R Þ B x = m I 4 (x 2 +R )
3/2 x • R r dB  θ B x = m IR 2
2(x +R )
3/2 B x m IR 2 2x 3
Note the form the field takes for x >> R:

16. (optional)

17. Circular Loop R B » x x B = m iR 2(x +R )
(optional) R B x 1 x 3 x B x = m iR 2
2(x +R )
3/2
Note: N (N small): N x Formula

18. Circ Loop Example
A circular coil of radius 5 cm has 12 turns and lies in the x=0 plane and is centered at the origin. It carries a 4 A ccw current. Find the magnetic field on the Ox axis at: a) x= 0, b) x= 15 cm, c) x = 3 m.
a) the center is N times the field for a single loop at the center:
b) B off axis is N times the field for a single loop off axis:
(optional)
(optional)
c) Use the previous equation again:
(optional)
Note: You can also use the version of the off axis equation for x>>R:

21. B-field of A Straight Wire
Calculate field at point P using Biot-Savart Law:
(Φ= θ+90)

22. B-field of ¥ Straight Wire
If wire stretches to infinity, θ2 and θ1 approach 90o and -90o respectively.
It results:
sin 90 = 1
sin (-90) = -1

25. Current I = 1.5 A, L = 0.5 m. Find the magnetic field in the center of the square.y
B = 4Bs P L I
L/2 O
Bs = 8.49 x 10-7 T
B = 4Bs = 3.39 x 10-6 T

26. 29.2: Magnetic Field due to a Current in a Circular Arc of Wire:

27. Example, Magnetic field at the center of a circular arc of a circle.:

28. Example, Magnetic field off to the side of two long straight currents:

29. 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

30. 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

31. Parallel Currents:
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

32. 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
1) 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.
I is up, B is out of the screen, so F is to the right.  the force is repulsive.

33. Questions (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

34. Questions (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

35. Questions (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

36. Questions (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!

37. 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

38. Next Time...
Currents source magnetic fields in several important examples
wires
solenoids
infinite sheet
Many cool examples!

39. 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

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