The figure shows a branching point
in an electric circuit. Find the current in branch 𝑥 and
the direction of the current in branch 𝑦.
Looking at our figure, we see
branches 𝑥 and 𝑦. Branch 𝑦 has a current of 10 amps
running through it. But we’re not sure whether that
current approaches or moves away from the node. And branch 𝑥 has a current we know
moves towards the branching point. But we don’t know the magnitude of
that current 𝐼. We want to solve for both 𝐼 as
well as the direction of the current in branch 𝑦.
To do all this, we’ll use a
principle of closed circuits sometimes called Kirchhoff’s node rule. What this node rule tells us is
that the sum of all current entering and leaving a branching point is zero. That means if we add up all the
current going into this branching point, that’s equal to all the current going out
of that point. Notice that this rule is
essentially a statement of the conservation of charge, that we can’t have charge
building up at any one point in the circuit.
Let’s apply this rule to our
particular branching point to solve for the magnitude of 𝐼 as well as the direction
of the current in branch 𝑦. As we do, we’ll adopt the
convention that current that’s leaving the branching point will have a negative
value to it and current that’s going into the branching point will have a positive
value to it. Here then is the equation we can
write when we consider all these currents going into and out of the branching
We can say that negative eight amps
plus two amps plus four amps plus nine amps plus or minus 10 amps, because we’re not
yet sure of the direction of the current in branch 𝑦, plus 𝐼 is all equal to
zero. If we combine together the four
current whose sign and magnitude we know, they all reduce down to seven amperes. So now, our equation reads: seven
amps plus or minus 10 amps plus 𝐼 is equal to zero.
Here is what we have then. We have two positive values, seven
amps and 𝐼, added together to a third value to give zero. That tells us that this third value
must be less than zero in order for the sum to be zero. So the 10-amp current is moving in
the negative direction, that is away from our branching point.
Now that we know the direction of
the current in branch 𝑦, we can solve for the magnitude 𝐼 by subtracting seven
amps from 10 amps. We find that the current 𝐼 is
equal to three amps. Here then is what we can say in
answer to our question. We can say that the current in
branch 𝑥 is three amps and the current in branch 𝑦 is away from the node. That’s how the current in the 𝑥
and 𝑦 branches behaves.