Interpreting Circuit Diagrams - Physics 2

1. From the circuit diagram, a student must determine the conventional current through $R_1$. Which of the following is correct?
   Answer: The current through $R_1$ is 2 A directed from P to Q, because the battery fixes the voltage across all parallel branches at 20 V, the ideal current source does not affect the terminal voltage since the battery is ideal, and $I_{R_1} = 20\ \text{V} / 10\ \Omega = 2\ \text{A}$. When a circuit contains both a battery and an ideal current source connected between the same two nodes, the first thing to ask is: what controls the node voltage? An ideal voltage source (your battery) enforces a fixed potential difference between its terminals regardless of what else is connected in parallel — this is the defining property of an ideal voltage source. Since the battery fixes $V_{PQ} = 20\ \text{V}$, every parallel branch sees exactly that voltage, full stop. With $V_{PQ} = 20\ \text{V}$ locked in, the current through $R_1$ follows directly from Ohm's Law: $I_{R_1} = 20\ \text{V} / 10\ \Omega = 2\ \text{A}$, directed from the high-potential node P toward Q. That confirms A is correct. B is wrong because it treats the current source as if it creates a voltage drop that "subtracts" from the battery. An ideal current source in parallel with an ideal voltage source cannot alter the node voltage — the battery overrides it. The current source simply adjusts how much current the battery itself must supply, not the terminal voltage. C is wrong on two counts: the direction is reversed without justification, and it misunderstands which element controls polarity. The battery's positive terminal at P keeps P at higher potential; current through resistors flows P → Q. D misapplies superposition. When you activate the current source alone (battery replaced by a short), the node voltage becomes zero (both nodes shorted), so the current source drives no additional current through $R_1$. Superposition gives the same 2 A result as A, not 2.5 A. Study tip: Whenever an ideal voltage source and any other element share the same two nodes, the voltage source wins — node voltage is fixed, and you apply Ohm's Law directly.
2. Which of the following statements about the student's KCL equations is correct?
   Answer: All three equations are correctly written, but they are not all independent — any one of the three equations can be derived from the other two, so only two of the three KCL equations can be used as independent constraints in solving the circuit. Whenever you see a question involving Kirchhoff's Current Law applied at every node of a circuit, you should immediately ask yourself: are all these equations truly independent? This is a fundamental property of KCL that's easy to overlook. Here's the key insight: in any circuit with $N$ nodes, applying KCL at every node produces exactly $N$ equations, but only $N-1$ of them are independent. Why? Because every branch current appears exactly twice across all node equations — once leaving a node (negative) and once entering another (positive). This means if you add all the KCL equations together, every current cancels, giving $0 = 0$. That redundancy proves one equation is always derivable from the others. You can verify this directly: adding the student's three equations gives $(-I_1 - I_3 + I_4) + (I_1 - I_2) + (I_2 + I_3 - I_4) = 0,$ which is $0 = 0$ identically. So the Node C equation is just the negative sum of Nodes A and B — it carries no new information. Answer B is correct. Answer A is wrong because it claims all three equations are independent, which contradicts this fundamental property. Only two can serve as independent constraints. Answer C is wrong because KCL is purely a current-conservation law — it counts charges flowing in and out of a node, regardless of whether a branch contains a resistor, battery, or anything else. Batteries don't get special treatment in KCL. Answer D is wrong because the student's sign convention is actually consistent: currents leaving Node A are negative, and $I_4$ enters Node A, so $+I_4$ is correct. Study tip: For any circuit with $N$ nodes, always use only $N-1$ KCL equations. The last one is always redundant — don't waste it as an independent equation in your system.

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