The figure shown represents a branch in an electric circuit. Find the current in branch 𝑋 and the direction of the current in branch 𝑌.
From the figure, we see that into this node, this branch, several currents are coming in and going out. On branch 𝑋, we see the direction of the current, but we don’t know its magnitude. That’s one thing we want to solve for. And on branch 𝑌, we see the magnitude, but not the direction of the current. That’s the second thing we want to solve for.
To help us solve for these values, we can recall the current branch rule, which says that the sum of current entering and leaving a node — also called a branch — is zero. This is essentially a statement of the conservation of charge. All the electrical charge moving into the node must be going out as well; otherwise, the charge will be accumulating somewhere.
Knowing this rule, let’s establish the convention that current flowing out of the node is negative and current flowing into the node has a positive sign to it. We can write then that negative eight amps plus two amps plus four amps plus nine amps plus or minus 10 amps, the current in branch 𝑌, plus 𝐼 is equal to zero. If we add up the first four current values listed, it comes out to seven amps, leaving us with the equation seven amps plus or minus 10 amps plus 𝐼 is equal to zero.
Knowing that the seven amp term as well as 𝐼 are both positive currents and that the sum of all three of these currents is equal to zero, that means the current in branch 𝑌 must be travelling in the negative direction. That’s the only way these three numbers can add up to zero. We can now solve this remaining equation for 𝐼, the current in branch 𝑋. 𝐼 is equal to 10 amps minus seven amps or three amps.
We can say then that the current in branch 𝑋 is three amps and the current in branch 𝑌 is away from the node. This is the current magnitude and direction we wanted to solve for.