1 Magnetic Field due to a Current-Carrying Wire Biot-Savart LawAP 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 WiresB = 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 straightWhat 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 VectorTo 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 LawThe 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 FieldMagnetic 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 WireIntegrating over all the current elements gives

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

12

13 B for a Curved Wire SegmentFind 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 WireConsider 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 LoopFind 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 ConductorsThe 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 LN: Number of turns L: Length n=N/L  L 

20 Direction of Force Between Two Parallel ConductorsIf 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, FBForce per unit length:

22 Definition of the AmpereWhen 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 CoulombThe 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 wiresDistance 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