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My question is that if we repeat the above procedure in vacuum, would the sparks be produced? Not in perfect vacuum, because there is no gas to ionize. An understanding of this behavior can be attained by studying Paschen's Law (see http://en.wikipedia.org/wiki/Paschen%27s_law), which describes the breakdown voltage for given distance and pressure.


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Next step in a systematic approach like you have started would be to use superposition. In that technique you find the contribution of each source separately to the $i_x$ and $V_x$. Be sure that you pay careful attention to all signs. A missed negative can really mess you up. First, find the (signed) contribution to $i_x$ and $V_x$ from the 5V supply by ...


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Yes, in general it affects the current, in particular if that current was going to change, the self inductance $L$ makes it change less quickly than it otherwise would change. Here are some examples: You could put a resistive loop inside a solenoid as a good example. The total $\vec{B}$ field is the $\vec{B}$ field due to the solenoid $\vec{B}_1(t)$, and ...



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