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Question

This is the circuit and the question is to find the voltage \$U_{J1}\$, if \$R_1, J_2, R_2, E, R_3\$ are known.

I solved it with four equations like this, but the end result isn't the same as in my book. Are these four equations correct?

Ohm's law for the resistor \$R_1\$ : $$U_{R_1}=J_1 R_1$$ Parallel branches that have the same voltage: $$U_{J_1}+U_{R_1}=U_{J_2}$$ Second Kirchoff's law for the central loop, where \$I_1\$ is the current in the branch with the resistor: \$R_2\$$$U_{J_2}+R_2I_1+E=0$$ First Kirchoff's law where we calculate \$I_1\$: $$I_1=J_1+J_2$$

Answer 1

The TLDR of this whole answer is: label everything, according to passive sign convention, where the higher potential (+) terminal of a resistor should be the one that labelled current enters.

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This seems to be a question about making simplifications, to avoid the need for a large number of simultaneous equations. The first simplification is to remove \$R_3\$, since it's in parallel with voltage source \$E\$. Any current through it, will flow via \$E\$ also, and not appear in any other equations, and it obviously cannot influence the potential difference \$E\$.

The second is to temporarily ignore \$R_1\$, since it is in series with \$J_1\$. Then replace the two current sources \$J_1\$ and \$J_2\$ with a single source \$J_1 + J_2\$. You already stated yourself that by KCL, current entering their bottom junction, and leaving the upper junction, will be \$J_1 + J_2\$, regardless of \$R_1\$.

What you are left with is this:

^{simulate this circuit – Schematic created using CircuitLab}

That's trivial to solve, permitting you to find the voltage \$U_{R2}\$ across \$R_2\$, and the potential difference \$U_{XY} = U_X - U_Y\$. What I have called \$U_{XY}\$ is actually your \$U_{J2}\$, but be very careful with polarities when you construct your KVL equation for this loop:

$$ U_{XY} - E - U_{R2} = 0 $$

Perhaps that's where you went wrong. Labels, and passive sign convention for resistors, always help. I've obviously got different signs from yours, but I don't have enough information from your question (or I'm too lazy to scrutinise your equations) to be sure that this was your error.

Then you can apply this new knowledge to the system of \$J_1\$, \$J_2\$ and \$R_1\$, always paying very careful attention to polarities:

^{simulate this circuit}

You have a complete expression for \$U_{XY}\$ (or \$U_{J2}\$), in terms of known quantities. This sub-system can then also be treated in isolation, using KVL to write an equation in terms of \$U_{XY}\$:

$$ U_{J1} - U_{R1} - U_{XY} = 0 $$

Maybe that's where you made a mistake, difficult to say without a full autopsy.