Another Application of Kirchhoff\'s Rules GOAL Find the currents in a circuit wi

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Another Application of Kirchhoff's Rules GOAL Find the currents in a circuit with three currents and two batteries when some current directions are chosen wrongly. PROBLEM Find I1, I2, and I3 in figure a. STRATEGY Use Kirchhoff's two rules, the junction rule once and the loop rule twice, to develop three equations for the three unknown currents. Solve the equations simultaneously. SOLUTION Apply Kirchhoff's junction rule to junction c. Because of the chosen current directions, I1 and I2 are directed into the junction and I3 is directed out of the junction. I3 = I1 + I2 Apply Kirchhoff's loop rule to the loops abcda and befcb. (Loop aefda gives no new information.) In loop befcb, a positive sign is obtained when the 6.0 Ohm resistor is traversed because the direction of the path is opposite the direction of the current I1. Loop abcda: 10 V - (6.0 Ohm) I1 - (2.0 Ohm) I3 = 0 Loop befcb: -(4.0 Ohm)I2 - 14 v + (6.0 Ohm) I1 - 10V = 0 Using Equation (1), eliminate I3 from Equation (2) (ignore units for the moment): 10 - 6I1 - 2.0 (I1 + I2) = 0 10 = 8.0I1 + 2.0I2 Divide each term in Equation (3) by 2 and rearrange the equation so that the currents are on the right side: -12 = -3.0I1 + 2.0I2 Subtracting Equation (5) from Equation (4) eliminates I2 and gives I1: 22 = 11I1 rightarrow I1 = 2.0 A Substituting this value of I1 into Equation (5) gives I2: 2.0I2 = 3.0I1 - 12 = 3.0(2.0) - 12 = -6.0 A I2 = -3.0 A Finally, substitute the values found for I1 and I2 into Equation (1) to obtain I3: I3 = I1 + I2 = 2.0 A - 3.0 A = -1.0 A LEARN MORE REMARKS The fact that I2 and I3 are both negative indicates that the wrong directions were chosen for these currents. Nonetheless, the magnitudes are correct. Choosing the right directions of the currents at the outset is unimportant because the equations are linear, and wrong choices result only in a minus sign in the answer. QUESTION Is it possible for the current through a battery to be directed from the positive terminal toward the negative terminal? (Select all that apply.) Yes. It happens inside a rechargeable flashlight battery or a car battery during recharge. Yes. It can happen when the voltage applied to a battery is greater than its emf. No. The charge carriers inside a battery is never reversible. PRACTICE IT Use the worked example above to help you solve this problem. Find Il, I2, and I3 in figure a, where epsilon = 15 V. I1 = I2 = I3 = Find the three currents in figure b, where R = 2.5 Ohm. (Note that the direction of one current may have been chosen wrongly!) I1 = There are many chances for algebraic errors in this problem. Go back and check your work carefully. A I2 = There are many chances for algebraic errors in this problem. Go back and check your work carefully. A I3 = There are many chances for algebraic errors in this problem. Go back and check your work carefully. A

Explanation / Answer

From figure :

I2=I1+I3

By kirchoff:

I1+I3+I3 = 0

2*I1+I3 = 0------(1)

I1+I3+2.5*I3 = 5

I1+3.5*I3 = 5------(2)

solving (1) and (2),we have

I1=-0.83333 A

I3= 1.666667

I2=0.83333 A

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