1. Magnetic Field due to a Current-Carrying Wire Biot-Savart Law
AP Physics C
Mrs. Coyle
Hans Christian Oersted, 1820

2. Magnetic fields are caused by currents.
Hans Christian Oersted in 1820’s showed that a current carrying wire deflects a compass.
Current in the Wire
No Current in the Wire

3. Right Hand Curl Rule

4. Magnetic Fields of Long Current-Carrying Wires
B = mo I
2p r
I = current through the wire (Amps)
r = distance from the wire (m)
mo = permeability of free space
= 4p x 10-7 T m / A
B = magnetic field strength (Tesla) I

5. Magnetic Field of a Current Carrying Wire

6. What if the current-carrying wire is not straight
What if the current-carrying wire is not straight? Use the Biot-Savart Law:
Assume a small segment of wire ds causing a field dB:
Note:
dB is perpendicular to ds and r

7. Biot-Savart Law allows us to calculate the Magnetic Field Vector
To find the total field, sum up the contributions from all the current elements I ds
The integral is over the entire current distribution

8. Note on Biot-Savart Law
The law is also valid for a current consisting of charges flowing through space
ds represents the length of a small segment of space in which the charges flow.
Example: electron beam in a TV set

9. Comparison of Magnetic to Electric Field
Magnetic Field
Electric Field
B proportional to r2
Vector
Perpendicular to FB , ds, r
Magnetic field lines have no beginning and no end; they form continuous circles
Biot-Savart Law
Ampere’s Law (where there is symmetry
E proportional to r2
Vector
Same direction as FE
Electric field lines begin on positive charges and end on negative charges
Coulomb’s Law
Gauss’s Law (where there is symmetry)

10. Derivation of B for a Long, Straight Current-Carrying Wire
Integrating over all the current elements gives

11. If the conductor is an infinitely long, straight wire, q1 = 0 and q2 = p
The field becomes: a

13. B for a Curved Wire Segment
Find the field at point O due to the wire segment A’ACC’:
B=0 due to AA’ and CC’
Due to the circular arc:
q=s/R, will be in radians

14. B at the Center of a Circular Loop of Wire
Consider the previous result, with q = 2p

15. Note
The overall shape of the magnetic field of the circular loop is similar to the magnetic field of a bar magnet.

16. B along the axis of a Circular Current Loop
Find B at point P
If x=0, B same as at center of a loop

17. If x is at a very large distance away from the loop.
x>>R:

18. Magnetic Force Between Two Parallel Conductors
The field B2 due to the current in wire 2 exerts a force on wire 1 of
F1 = I1ℓ B2

19. Magnetic Field at Center of a Solenoid B = mo NI L
N: Number of turns
L: Length
n=N/L
 L 

20. Direction of Force Between Two Parallel Conductors
If the currents are in the:
same direction the wires attract each other.
opposite directions the wires repel each other.

21. Magnetic Force Between Two Parallel Conductors, FB
Force per unit length:

22. Definition of the Ampere
When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

23. Definition of the Coulomb
The SI unit of charge, the coulomb, is defined in terms of the ampere
When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C

24. Biot-Savart Law: Field produced by current carrying wires
Distance a from long straight wire
Centre of a wire loop radius R
Centre of a tight Wire Coil with N turns
Force between two wires