21pt in the circuit shown and the batteries have emfs

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Unformatted text preview: circuit shown, and . The batteries have emfs of and . (a) [6pt] Write Kirchoff’s Loop laws for the upper and lower loops in the figure. Express them in terms of symbols in the figure. • • The sum of potential differences around each loop is zero: , (b) [3pt] Write the expression resulting from applying Kirchoff’s Junction Law in terms of symbols in the figure. The total current going into either junction is zero, so (c) [12pt] Find the currents and numerically. Show your work and make your final answers clear by putting them in boxes, with correct units. First, eliminate from the second loop equation: The two loop equations can now be written in terms of just two currents: Solving the first of these for gives Substituting this into the second equation gives 4 Combining like terms in this expression gives Therefore, Substituting this into the earlier expressions gives Note: A completely symbolic solution would lead to Alternative solution: The currents depend linearly on the emfs of the batteries, so you can calculate the currents by replacing one of the batteries at a time by a wire (“shorting” it, so that calculating the currents with just a single battery and a combination of resistors, and then adding the currents for each case. This is the “method of superposition.” Shorting attached to gives an equivalent resistance of so , and the current divider for gives (Note the arrow directions. The actual current direction through Shorting attached to is to the right in this case.) gives an equivalent resistance of so and the current divider for gives Adding the currents for the two solutions gives The results are consistent up to small rounding differences in the last digit, but one more digit is shown here than is actually justified, so the solutions agree. 5 Part 3. [15...
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This document was uploaded on 03/09/2014 for the course PHYS 222 at Ohio Northern University.

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