Answer
$V_{ab}=-8V$
$P_{I_1}=-24W$
$P_{I_2}=8W$
$P_{R_1}=5.33W$
$P_{R_2}=10.67W$
Work Step by Step
We first apply Kirchhoff's Current Law and Ohm's law, which states that $I=\frac{V}{R}$, to find:
$3-1+\frac{V_{ab}}{12} + \frac{V_{ab}}{6} = 0 \\ 2 + \frac{3V_{ab}}{12} = 0 \\ 3V_{ab} = -24 \\ V_{ab} = \fbox{-8V} $
We use the equation for power, recalling that if the current enters through the negative terminal, a negative sign is added in front of $VI$ in the equation. Thus, we find:
$P_{I_1}=VI = -8 \times 3 = \fbox{-24W}$
$P_{I_2}=-VI = -(-8 \times 1) = \fbox{8W}$
$P_{R_1}= \frac{V^2}{R} = \frac{8^2}{12} = \fbox{5.33W}$
$P_{R_2}= \frac{V^2}{R} = \frac{8^2}{6} = \fbox{10.67W}$
Note, if the power is positive, the element is absorbing energy, while if the power is negative, the element is giving off energy.
