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

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

Right Hand Curl Rule

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

Magnetic Field of a Current Carrying Wire http://www.walter-fendt.de/ph14e/mfwire.htm

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

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

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

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)

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

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

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

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

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

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

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

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

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

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.

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

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

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

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