4 Magnetic Fields of Long Current-Carrying Wires B = mo I2p rI = current through the wire (Amps)r = distance from the wire (m)mo = permeability of free space= 4p x 10-7 T m / AB = magnetic field strength (Tesla)I
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 dsThe 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 spaceds 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 FieldElectric FieldB proportional to r2VectorPerpendicular to FB , ds, rMagnetic field lines have no beginning and no end; they form continuous circlesBiot-Savart LawAmpere’s Law (where there is symmetryE proportional to r2VectorSame direction as FEElectric field lines begin on positive charges and end on negative chargesCoulomb’s LawGauss’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 NoteThe 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 PIf 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 ofF1 = I1ℓ B2
19 Magnetic Field at Center of a Solenoid B = mo NI L N: Number of turnsL: Lengthn=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 ampereWhen 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 wireCentre of a wire loop radius RCentre of a tight Wire Coil with N turnsForce between two wires
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