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The Curl of a Magnetic Field

The Curl of a Magnetic Field - point We have already...

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The Curl of a Magnetic Field From this equation, we can generate an expression for the curl of a magnetic field. Stokes' Theorem states that: B · ds = curl B · da We have already established that B · ds = . Thus: curl B · da = To remove the integral from this equation we include the concept of current density, J . Recall that I = J · da . Substituting this into our equation, we find that curl B · da = J · da Clearly, then: = Thus the curl of a magnetic field at any point is equal to the current density at that point. This is the simplest statement relating the magnetic field and moving charges. It is mathematically equivalent to the line integral equation we developed before, but is easier to work with in a theoretical sense. The Divergence of the Magnetic Field Recall that the divergence of the electric field was equal to the total charge density at a given
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Unformatted text preview: point. We have already examined qualitatively that there is no such thing as magnetic charge. All magnetic fields are, in essence, created by moving charges, not by static ones. Thus, because there are no magnetic charges, there is no divergence in a magnetic field: = 0 This fact remains true for any point in any magnetic field. Our expressions for divergence and curl of a magnetic field are sufficient to describe uniquely any magnetic field from the current density in the field. The equations for divergence and curl are extremely powerful; taken together with the equations for the divergence and curl for the electric field, they are said to encompass mathematically the entire study of electricity and magnetism....
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