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Unformatted text preview: circuit shown,
. The batteries have
(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.”
attached to gives an equivalent resistance of
, and the current divider for gives (Note the arrow directions. The actual current
attached to is to the right in this case.) gives an equivalent resistance of
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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