Questions: DC Circuit Analysis with Kirchhoff's Laws

Question 1

At a circuit node, three branches meet. Branch 1 carries 3 A flowing into the node, Branch 2 carries 5 A flowing into the node, and Branch 3's current is unknown. What is Branch 3's current, and which direction does it flow?

• A. 2 A, flowing into the node
• B. 8 A, flowing out of the node
• C. 8 A, flowing into the node
• D. 2 A, flowing out of the node

Answer: (B)

KCL states that the sum of currents into a node equals the sum out: 3 + 5 = I₃. Therefore I₃ = 8 A flowing out of the node. This follows from charge conservation — charge cannot accumulate at a node in steady state. Option A incorrectly subtracts; option C would violate charge conservation by having all three currents flow in. The direction (out of the node) is determined by the requirement that inflow equals outflow.

Question 2

A student applies KVL to a loop containing a 12 V battery and two resistors (4 Ω and 8 Ω) in series, with all current flowing clockwise. Traversing the loop clockwise, they record +12 V for the battery. What should they record for each resistor?

• A. +IR for each resistor, since current flows clockwise and so does the traversal
• B. −IR for each resistor, since crossing a resistor with the current direction is a voltage drop
• C. +IR for one resistor and −IR for the other, depending on resistor size
• D. 0 V for each resistor, since resistors don't affect voltage in a series loop

Answer: (B)

When traversing a resistor in the direction of conventional current flow, you experience a voltage drop: the voltage decreases by IR, so you record −IR. Since current flows clockwise and the traversal is also clockwise, both resistors are crossed with the current direction, giving −I(4) and −I(8). KVL then gives: 12 − 4I − 8I = 0, so I = 1 A. The sign convention is: crossing a resistor with the current = voltage drop (−IR); crossing against the current = voltage rise (+IR).

Question 3

Kirchhoff's Voltage Law states that the sum of currents entering a node equals the sum of currents leaving that node.

• T. True
• F. False

Answer: False

This is Kirchhoff's Current Law (KCL), not KVL. Kirchhoff's Voltage Law states that the sum of all voltage changes around any closed loop is zero: ∮V = 0. KVL follows from the conservative nature of the electrostatic field — if you start and end at the same point, the net change in electric potential must be zero. Confusing KCL with KVL is a common error; remember: KCL is about currents at a node; KVL is about voltages around a loop.

Question 4

Kirchhoff's laws are approximations that work well for simple circuits but break down for complex networks with many branches.

• T. True
• F. False

Answer: False

Kirchhoff's laws are exact consequences of Maxwell's equations in the low-frequency (lumped-circuit) approximation — they are not approximations themselves. KCL follows exactly from charge conservation; KVL follows exactly from the conservative nature of the electrostatic field. They apply to arbitrarily complex networks — indeed, the node voltage and mesh current methods use them to solve systems with dozens of unknowns. The only regime where Kirchhoff's laws break down is at very high frequencies where the lumped-circuit approximation fails and electromagnetic wave effects become significant.

Question 5

Explain in your own words why KVL guarantees that the sum of voltages around any closed loop is zero. What physical principle underlies this?

Model answer: KVL follows from the fact that the electric field in a circuit is conservative. A conservative field means that the work done moving a charge from point A back to point A along any closed path is zero — the path doesn't matter, only the endpoints, and if you end where you started, the net energy change is zero. Voltage is electric potential energy per unit charge, so traversing a closed loop and returning to the starting point must yield zero net change in voltage. Every element either adds potential (like a battery) or drops it (like a resistor), and these must exactly cancel around any closed loop.

Grounding KVL in the conservative nature of the electrostatic field makes it clear why it is an exact law rather than an empirical rule. This understanding also explains why KVL holds for any closed loop you choose to draw through a circuit — not just the 'obvious' ones — which is what makes the mesh current method so powerful.