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Biot-Savart Law

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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  1. Biot-Savart Law permeability of free space B B

  2. Magnetic Field Surrounding a Thin, Straight Conductor If the wire is very long, then

  3. Magnetic Field on the Axis of a Circular Current Carrying Loop At O (x=0) At x>>R Special Cases:

  4. Field Due to a Circular Arc of Wire Full Circle (f = 2p)

  5. 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

  6. 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

  7. 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

  8. 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:

  9. 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

  10. 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.

  11. 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:

  12. Ex. 30.8Magnetic Flux Through a Rectangular Loop area element dA = b dr. Because r is the only variable  dA Wire

  13. 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

  14. 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

  15. 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)

  16. For quarter circle 1/4 B-field of full circle Or, B-field due to circular curve is

  17. 16 F2=FB= I2lB1FB/l=I1B1= (8A)(1×10-5 T) = 8×10-5 N/m downward (b) (c) (d) F1=FB= I1lB2FB/l=I1B1= (5A)(1.6×10-5 T) = 8×10-5 N/m upward

  18. FB What forces affect the proton? 1) mg downward 2)FB upward mg mg = FB mg = qυB , but B = μ0I/2πd  mg = qυμ0I/2πd

  19. (a) A (surface vector) (b)

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