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Biot-Savart Law. permeability of free space. B. B. Magnetic Field Surrounding a Thin, Straight Conductor. If the wire is very long,. then. Magnetic Field on the Axis of a Circular Current Carrying Loop. At O (x=0). At x>>R. Special Cases:. Field Due to a Circular Arc of Wire.

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## Biot-Savart Law

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**Biot-Savart Law**permeability of free space B B**Magnetic Field Surrounding a Thin, Straight Conductor**If the wire is very long, then**Magnetic Field on the Axis of a Circular Current Carrying**Loop At O (x=0) At x>>R Special Cases:**Field Due to a Circular Arc of Wire**Full Circle (f = 2p)**30.2 Magnetic Force Between Two Parallel Conductors**FB between two parallel wires Opposite in direction I1& I2 same in direction attraction I1& I2 opposite in direction repulsion Force per unit length If a = 1m, I1 = I2, and FB/l = 2×10-7N/m I in both wires is defined to be 1 ampere**30.3 Ampère’s Law**I=0 no B-field (a) With I ≠ 0 B-filed Loop (b) Infinite wire Integrate around the loop A line integral of B.ds around a closed path equals m0I, whereIis the total continuous current passing through any surface bounded by the closed path (amperian loop). Ampere’s law**Ex: 30.4 Long Current Carrying Wire**For r >= R , I0pass through whole surface Calculate the B-field a distance r from the center of the wire in the regions r >= R and r < R. For r < R amperian loops**Ex: 30.5 The Toroid**For a toroid having N closely spaced turns of wire, calculate the B-field in the region occupied by the torus, a distance r from the center. By symmetry, B is constant over the dashed circle and tangent to it Outside the toroid:**30.4 The B-Field of a Solenoid**A solenoid is a long wire wound in the form of a helix uniform B-field in the interior net B-field is the vector sum of the fields resulting from all the turns. Almost uniform B-field in the interior**The Magnetic Field of a Solenoid**Consider long solenoid L>> R Along path 2 and 4, (B┴ds) B.ds= 0 Along path 3, B=0 where n is the number of turns per unit length.**30.5 Magnetic Flux (ΦB)**The defenition of ΦB is similar to the electric flux ΦB. If we have element area dA with magnetic filed B passing through it, then dA is the Surface vector (Weber=Wb=T.m2) For a uniform field making an angle q with the surface normal:**Ex. 30.8Magnetic Flux Through a Rectangular Loop**area element dA = b dr. Because r is the only variable dA Wire**30.6 Gauss’ Law in Magnetism**Unlike electrical fields, all magnetic field lines always form loops. (always here is a dipole). Hence, Net flux over any closed surface equal to zero number of line entering = number of lines leaving Electric Field Lines Magnetic Field Lines**Summary**Ampère’s law Biot–Savart law the magnitude of the magnetic field at a distance r from a long, straight wire carrying an electric current I is Total B-filed force per unit length between two parallel wires separated by a distance a is The magnitudes of the fields inside a toroid The magnitudes of the fields inside a solenoid B-field due to a circular Arc of Wire magnetic flux ΦBthrough a surface Net magnetic flux ΦBover a closed surface is zero B-field due to a full circle**υ**2.19×106 m/s R=5×10-11 m Discussion Ch.30 (1, 7, 16, 22, 31, 35) B-field at the center of the circle is (Ex. 30.3): But, I = q/t , t =distance/speed= 2πR/υ I = q(υ/2πR)**For quarter circle**1/4 B-field of full circle Or, B-field due to circular curve is**16**F2=FB= I2lB1FB/l=I1B1= (8A)(1×10-5 T) = 8×10-5 N/m downward (b) (c) (d) F1=FB= I1lB2FB/l=I1B1= (5A)(1.6×10-5 T) = 8×10-5 N/m upward**FB**What forces affect the proton? 1) mg downward 2)FB upward mg mg = FB mg = qυB , but B = μ0I/2πd mg = qυμ0I/2πd**(a)**A (surface vector) (b)

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