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

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

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Right Hand Curl Rule

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

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

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

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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

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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

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Comparison of Magnetic to Electric Field Magnetic Field B proportional to r 2 Vector Perpendicular to F B, 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 Electric Field E proportional to r 2 Vector Same direction as F E Electric field lines begin on positive charges and end on negative charges Coulomb’s Law Gauss’s Law (where there is symmetry)

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

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

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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: s/R, will be in radians

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

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

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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

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

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Magnetic Force Between Two Parallel Conductors The field B 2 due to the current in wire 2 exerts a force on wire 1 of F 1 = I 1 ℓ B 2

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Magnetic Field at Center of a Solenoid B = o NI L N: Number of turns L: Length n=N/L L

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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.

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Magnetic Force Between Two Parallel Conductors, F B Force per unit length:

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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 N/m, the current in each wire is defined to be 1 A

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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

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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

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