How many currents are there in a circuit?

Question

I've been studying circuits lately and I have a difficulty when applying Kirchhoff's rules. My question has to do with how can I find how many currents are in a circuit. I know about junctions etc and how it splits. But for example, when you have 2 or 3 batteries how many currents are there and which current runs through each resistor? I've been trying to practice online and I came across this circuit:

And I don't understand why there are 3 currents when there are 4 batteries. Can anyone help me please?

Answer 1

How many currents there are has nothing to do with how many batteries there are. Each uninterrupted part of the circuit will have its own amount of current going through it. If two batteries are in that same part, they will carry the same current ($E_3$ and $E_4$ in your case).

An uninterrupted part in this case means any stretch of circuit that does not have any way for current to leave (like a junction).

Instead of current, you can also think of it as a liquid going through pipes to make the intuition easier.

Answer 2

In this type of circuit, the number of uniquely defined currents can be equal to the number of “holes” through the circuit. In this case there are two holes and $I_1$ can be expressed as ${I_2} + {I_3}$. I like to work with current loops and voltage drops. In this case, if $I_1$ and $I_2$ each loop around their corresponding holes, then the current down through $R_3$ is ${I_1} – {I_2}$.

Answer 3

The by the book approach is to say there are 8 wire segments (between two nodes) each with its own current and voltage equation. So we start with 8 currents.

For example $V_{\rm bc} = -R_3 I_{\rm bc} $ or $V_{\rm ge} = E_2 - r_2 I_{\rm ge}$.

Now there is a reduction in variables that can happen for each node that connects two wires only. The current is shared between the connected wires, so $I_2 = I_{\rm bc}= I_{\rm cd}= I_{\rm de}$ for example. This leaves us with 3 currents.

But the circuit can be described by two loops, with some wires shard between the loops. These two loop currents $I_A$ and $I_B$ are used with the principal of superposition to describe the current on each wire.

For example $V_{\rm ge} = E_2 - r_2 (I_A-I_B)$

In this sense there are only 2 currents.