Phys4W09HmwkSolutions3 - Physics 4 Winter 09 Homework...

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Physics 4 Winter 09 Homework Assignment 3 Mike Blume January 27, 2009 1 Amp` ere’s Law Explained Amp` ere’s Law is often written H ~ B ( ~ r ) · d ~ l = μ 0 I encl Part A The integral on the left is: A the integral throughout the chosen volume. B the surface integral over the open surface. C the surface integral over the closed surface bounded by the loop. D the line integral along the closed loop. E the line integral from start to finish. solution Answer: D We can eliminate our first three answers by simply looking at the differ- ential d ~ l . This indicates that the integral is taken over a length, not a volume or an area. E suggests that the integral in question has a well-defined be- ginning and end, but the symbol H instead implies an integral over a closed loop. Thus we go with D. Part B What physical property does the symbol I encl represent? A The current along the path in the same direction as the magnetic field B The current in the path in the opposite direction from the magnetic field C The total current passing through the loop in either direction D The net current through the loop 1
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solution In this case, the subscript ‘encl’ is intended to mean ‘enclosed’. Thus, we must concern ourselves with the space within the loop. This rules out A and B. On the other hand, C is nearly nonsensical. Imagine a chunk of ordinary matter passing through the loop, say for instance your hand. Your hand contains many moles of positively charged protons, as well as many moles of negatively charged electrons. Thus, as your hand goes through, all those protons could be thought of as a current going one way, and the electrons could be thought of as a current going in the opposite way. If we take Amp` ere’s law in the sense of part C, these currents add. Let’s do a quick back of the envelope calculation here. The charge of an electron carries a factor of 10 - 19 . Avogadro’s Number carries a factor of 10 26 . This means that a mole of electrons is up there in the megacoulombs. If C were true, you could produce unimaginably large magnetic fields simply by waving your hands about. This would certainly save the folks at the LHC a lot of trouble, but would be problematic for everyday life. In any case, it is not what we observe. We are left with answer D. If we take the net current, then the current from all the protons in your hand cancels the current from all the electrons in your hand, and you do not produce world-shattering magnetic fields simply by waving. Part C The circle on the integral means that ~ B ( ~ r ) must be integrated A over a circle or a sphere. B along any closed line that you choose. C along the path of a closed physical conductor. D over the surface bounded by the current-carrying wire. solution Well, this one’s pretty straightforward. It is a fact of mathematical notation that the circle means any closed path. That leaves us with B and C. Now, if you took a path along a conductor, you wouldn’t really be enclosing current.
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