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Systems have mutual inductance and self inductance-however won't the effects of these two effect each other as well? Do systems undergo current and magnetic field behavior that are, in this way, dictated by differential equations?

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In principle you can add some linear coupling and solve coupled differential equations which would be functions of time, but once the system is in steady state, the effects vanish. You can have a look at the problem solving tutorial at UMD webpage

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In the steady state situation, what are you left with? Just the two magnetic fields from the currents? (Does that equation have a name?) – user24082 May 12 '13 at 17:42

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