# Why can we assume a current direction when using kirchhoff's circuit law?

Reference direction: When analyzing electrical circuits, the actual direction of current through a specific circuit element is usually unknown. Consequently, each circuit element is assigned a current variable with an arbitrarily chosen reference direction. When the circuit is solved, the circuit element currents may have positive or negative values. A negative value means that the actual direction of current through that circuit element is opposite that of the chosen reference direction.

Why can we assume a direction and get the correct value and sign? Is there a simple proof of this fact?

• Don't understand your question. You assume certain directions and then you get the correct answer values, with those answer values including the correct signs. – Samuel Weir Dec 10 '16 at 3:53
• @SamuelWeir Thank you for pointing out. I've edited my question. – chen Dec 10 '16 at 7:12

Is there a simple proof of this fact?

Sure; when you insert an ammeter into a circuit branch, there are two choices of polarity which amount to choosing a reference direction.

If two identical ammeters, connected in series with opposite polarity, are inserted into a circuit branch, they will measure the same current but give the opposite sign since each has a different reference direction.

However, they both give the same information. For one ammeter, current enters the positive lead and this ammeter gives a positive reading. For the other ammeter, current exits the positive lead and this ammeter gives a negative reading.

In either case, the ammeter reading gives you the correct direction of the current.

• Thank you for your answer. Do you know if there is a mathematical proof of this? – chen Dec 10 '16 at 5:41
• @chen- A mathematical proof for a circuit network is also easy. Usually you get a system of linear equations to determine the currents in the circuit branches after you assumed the positive current directions in the branches. Thus your solution will be $I_1, I_2,...,I_k,...I_n$. If you change the current arrow in the kth branch, you solution will be $I_1', I_2',...,I_k',...I_n'$, where $I_k'=-I_k$, giving you the same physical current $I_k$ as before. You can continue this with arrow direction changes in arbitrary loops. – freecharly Dec 10 '16 at 8:04
• @chen, the point of my answer is that a mathematical proof is unnecessary; picking a reference direction is physically equivalent to choosing which orientation to insert your ammeter. Put another way a positive current 'to the right' is identical to a negative current 'to the left'. It's the same with, e.g., velocity; 60mph east is the same as -60mph west. The difference in sign is due the choice of reference direction. – Alfred Centauri Dec 10 '16 at 13:22

You can assume arbitrary but then fixed current directions in the circuit branches of a circuit because the result just changes sign when you change the chosen direction. Thus if you change in one branch $n$ the assumed positive current direction your calculation result for the current $I_n$ changes to -$I_n$ so that the actual physical current value and orientation stays the same. It is similar to the case when you consider a body with velocity $\vec v$ in positive x-direction and then make the coordinate transform $x \to -x$. The body still moves in the same direction but the velocity is $-v$.

Let's assume you have a one loop circuit, a battery and a resistor connected to it. Now you chose an arbitrary direction (draw an arrow) in the resistor and say that a current flowing in this arrows direction will be called a positive current. If you draw your arrow going from the plus to the minus terminal of the battery, you will have a positive current, because charge flows through the resistor from plus to minus. If you draw your arrow in the opposite direction (from minus to plus) you will have a negative current in that direction meaning that you have a positive current from plus to minus as before when you described the same situation with the arrow from plus to minus.

• "You can assume arbitrary but then fixed current directions in the circuit branches of a circuit because the result just changes sign when you change the chosen direction." Could you explain more about the reason behind this? Or could you suggest further reading on this? Thank you. – chen Dec 10 '16 at 7:21
• @Chen - I add an explanation to my answer. – freecharly Dec 10 '16 at 7:47