The Biot-Savart Law with Applications.

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Field of a Straight Wire at an Off-Axis Point P Y a wire L 1 I L 2 X We want to find the magnetic field at the location P (0,a) created by the current in a straight wire located along the X axis from L 1 to L 2 . P a Y r = (x 2 + a 2 ) 1/2 r^ 180-θ θ ds X L x 1 L 2 Step 1: We find the field caused by the current in ds. Using the RHR: r d s × rˆ = kˆ ( dssinθ ) We replace ds by dx and sinθ by: sin ( θ ) = sin( 180−θ) = a r r dB = µ 0 4π v Ids × rˆ r 2 ˆ µ 0Idx ⎛ a = k 2 ⎜ 4πr ⎝ r ⎞ ⎟ ⎠ ˆ µ 0Iadx = k 3 4πr = kˆ 4π µ Iadx 0 3 2 2 ( x + a )2 4
B We integrate dB over the limits x = L 1 to x = L 2 : L r 2 ˆ µ 0Ia = ∫ dB = k 4π ∫ L r ⎡ = ˆ µ 0I B k ⎢ 4πa ⎢ ⎣ 1 dx ˆ µ 0Ia ⎡ = k ⎢ ( ) 3 4π 2 2 ⎢ 2 2 2 x + a 2 ⎣a ( x + a ) L − 2 ( + ) 1 ( + ) 1 ⎥⎥ 2 2 2 2 L a L1 a ⎦ 2 2 2 Usually it is more convenient to obtain the magnitude of this field from the following equation and obtain the correct direction from a new right hand rule shown on the next slide. L 1 ⎤ x 1 2 ⎤ ⎥ ⎥ ⎦ L L 2 1 µ 0I B = 4 π a L 1 2 ( ) 1 2 2 2 L + a ( L + a ) 2 2 2 2 − 1 L 1 B B These diagrams shown end views of two straight wires. The top wire has a current directed out of the page while the bottom wire has a current into the page. The magnetic field is circular about each wire. The direction is easily obtained from a new RHR of convenience. X Imagine grasping the wire with your right hand with your thumb extended in the current direction. Your fingers will wrap around the wire in the direction of the circular magnetic field. 5
Page 1 and 2: wire I dB r θ ds r P The objective
Page 3: Y θ = π/2 ds Step 2: Integrate to

The Biot-Savart Law with Applications.

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