Cracking the AP Physics B Exam, 2014 Edition by Princeton Review

Author:Princeton Review [Review, The Princeton]
Language: eng
Format: epub
ISBN: 978-0-8041-2423-2
Publisher: Random House Children's Books
Published: 2013-10-14T16:00:00+00:00

Now pick a loop; say, abcfa. Starting at a, we go to b, then across R1 in the direction of the current, so the potential drops by I1R1. Then we move to c, then up through R2 in the direction of the current, so the potential drops by (I1 − I2)R2. Then we reach f, turn left and travel through V1 from the negative to the positive terminal, so the potential increases by V1. We now find ourselves back at a. By the Loop Rule, the total change in potential around this closed loop must be zero, and

–I1R1 − (I1 − I2)R2 + V1 = 0 (1)

Since we have two unknowns (I1 and I2), we need two equations, so now pick another loop; let’s choose cdefc. From c to d, we travel across the resistor in the direction of the current, so the potential drops by I2R3. From e to f, we travel through V2 from the positive to the negative terminal, so the potential drops by V2. Heading down from f to c, we travel across R2 but in the direction opposite to the current, so the potential increases by (I1 − I2)R2. At c, our loop is completed, so

–I2R3 − V2 + (I1 − I2)R2 = 0 (2)

Substituting in the given numerical values for R1, R2, R3, V1, and V2, and simplifying, these two equations become

3I1 − 2I2 = 2 (1´)

2I1 − 5I2 = 1 (2´)

Solving this pair of simultaneous equations, we get

I1 = A = 0.73 A and I2 = A = 0.09 A

So the current through R2 is I1 − I2 = A = 0.64 A.

The choice of directions of the currents at the beginning of the solution was arbitrary. Don’t worry about trying to guess the actual direction of the current in a particular branch. Just pick a direction, stick with it, and obey the Junction Rule. At the end, when you solve for the values of the branch current, a negative value will alert you that the direction of the current is actually opposite to the direction you originally chose for it in your diagram.

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