Science:Math Exam Resources/Courses/MATH152/April 2022/Question B5 (a)/Solution 1

We are given the two loops to which the currents ${\textstyle i_{1}}$ and ${\textstyle i_{2}}$ correspond, and we can use Kirchoff’s voltage law on these loops to determine equations for ${\textstyle i_{1}}$, ${\textstyle i_{2}}$, and ${\textstyle E}$.

In the left loop: ${\textstyle -V_{1}+i_{1}\cdot 1+-V_{2}=0}$

In the right loop: ${\textstyle V_{2}+i_{2}\cdot 2-E=0}$

And, we also know that if there is a current source in a loop, the loop current must equal the current source, so we have ${\textstyle i_{2}=I}$.

Rearranging gives us

$${\begin{aligned}i_{1}&=V_{1}+V_{2}\\i_{2}&=I\\E&=2I+V_{2}\end{aligned}}.$$