Chapter 8 - The Steady Magnetic Field

Chapter 8 - The Steady Magnetic Field - Chapter 8 The...

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Chapter 8 The Steady Magnetic Field
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Sources of Magnetic Fields 1. Permanent magnet 2. Linearly-changing electric field 3. Direct current
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Biot-Savart Law At any point P, the magnitude of the magnetic field intensity produced by a differential current element is proportional to the product of the current , the magnitude of the differential length , and the sine of the angle lying between the filament and the line connecting the filament to P. The magnitude of the magnetic field is inversely proportional to the square of the distance from the differential element to P . The direction of the magnetic field intensity is normal to the plane containing the differential element and the line drawn from the filament to P. Of the two normals, the one in the direction of progress of a right-handed screw turned from the direction of current is chosen. The constant of proportionality is 1/4 π . 3 2 R 4 d I R d I 4 1 d π × = × π = R L a L H r
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If the differential current is at point 1 and the point where the magnetic field intensity is desired is point 2, then 12 2 1 R 4 d I d π × = R12 a L H 1 2
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To find the magnetic field due to a closed circuit, integrate the equation over the entire circuit (current path): π × = 2 R 4 Id r a L H
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For distributed sources such as current density J and surface current density K , then, I d L = K dS = J dv, so: π × = S 2 R 4 dS r a K H and π × = vol 2 R 4 dv r a J H
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Example: An infinitely long straight conducting filament at the z-axis is carrying a direct current I directed along +z direction. Determine the magnetic field intensity at a point P on the xy plane that is ρ units away from the z-axis.
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R = ρ a ρ - z'a z d L = dz’ a z a r = d H = 2 2 ' z ' z + ρ - ρ z ρ a a 2 / 3 2 2 ) ' z ( 4 ) ' z ( ' Idz + ρ π - ρ × z ρ z a a a 3 2 R 4 Id R Id 4 1 d π × = × π = R L a L H r
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φ - 2 φ - φ - πρ = + ρ ρ π ρ = + ρ ρ π = + ρ π - ρ × = a
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Chapter 8 - The Steady Magnetic Field - Chapter 8 The...

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