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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. Current in the Wire No Current in the Wire

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Right Hand Curl Rule

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**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

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**Magnetic Field of a Current Carrying Wire**

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**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

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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 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)

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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, q1 = 0 and q2 = p**

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: q=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 q = 2p

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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 B2 due to the current in wire 2 exerts a force on wire 1 of F1 = I1ℓ B2

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**Magnetic Field at Center of a Solenoid B = mo 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, FB**

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 10-7 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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