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# lecture20 - Sour ces of M a gneti c Fi el ds Sour B esi des...

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Sources of Magnetic Fields Sources of Magnetic Fields

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Besides magnetic poles, electric currents create magnetic fields. There are two ways of calculating B produced by currents: i) Biot-Savart Law : Field of a “current element” (analogous to a point charge in electrostatics). i) Ampère’s Law : An integral theorem similar to Gauss’s law.
m/A T 10 4 7 o × = - π μ ˆ 4 2 o r I r ds dB × = π μ We get the total B by integrating ds along the wire : (“permeability of vacuum”) I I θ ds r r dB ˆ 4 2 o × = r I r ds B π μ Biot-Savart Law Biot-Savart Law

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Question: When is d B =0 ? How does d B vary with r ? Current I in a straight wire What is the direction of B at points a and b ? a b
Example 1 Example 1 Find B at a distance R from a long straight wire x y I R φ dx dB θ i ds ˆ = dx direction) - z (in o dx r I dB 2 sin 4 θ π μ = (note sin = R/ r = cos θ φ ) r

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Solution
Result : the magnitude of the field produced at a distance R from a long straight wire is: R I B π μ 2 0 = I I The field lines form circles around the wire. Note the

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lecture20 - Sour ces of M a gneti c Fi el ds Sour B esi des...

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