Questions: Parallel Circuits: Conductance and Current Division

Question 1

Two resistors R₁ = 2Ω and R₂ = 8Ω are connected in parallel across a voltage source. What fraction of the total current flows through R₁?

• A. 1/5, because R₁ is one of two resistors and takes the smaller share
• B. 2/10 = 1/5, computed as R₁/(R₁ + R₂)
• C. 4/5, computed as R₂/(R₁ + R₂) — the smaller resistor gets the larger share
• D. 1/2, because both resistors receive the same voltage and current splits equally

Answer: (C)

The current divider formula gives I₁/I_total = G₁/(G₁ + G₂) = R₂/(R₁ + R₂) = 8/(2+8) = 4/5. This counter-intuitive form (current through R₁ involves R₂ in the numerator) trips beginners because it seems backward — but the logic is that current preferentially takes the path of least resistance. R₁ = 2Ω is the lower-resistance path, so it carries the larger share (4/5) of the current. Option B is the common error: writing R₁/(R₁+R₂) = 2/10 = 1/5, which gives the wrong branch.

Question 2

Why does connecting more resistors in parallel always decrease the total resistance of the network?

• A. Because each additional branch carries some current away from the others, reducing overall current flow
• B. Because total resistance is the average of the branch resistances, and averaging more values pulls the result down
• C. Because each additional branch provides another pathway for current, increasing total conductance and thus decreasing total resistance
• D. Because parallel resistors must share the same current, forcing each to work less hard

Answer: (C)

In parallel, conductances add: G_total = G₁ + G₂ + .... Each additional branch adds a positive conductance, so G_total strictly increases, meaning R_total = 1/G_total strictly decreases. Conceptually, each new branch opens another route for current to flow, making the combined network easier to drive with the same voltage. This is why total parallel resistance is always less than the smallest individual resistance — adding even a high-resistance branch still adds some conductance. Option A has it backwards: parallel branches each carry more total current from the source, not less.

Question 3

In a parallel circuit, the voltage across every branch is the same, regardless of the resistance of each branch.

• T. True
• F. False

Answer: True

This follows directly from Kirchhoff's loop rule. All branches in a parallel group share the same two nodes. By the loop rule, the potential difference between those two nodes must be the same regardless of which path you trace between them. This equal-voltage property is the defining characteristic of parallel circuits — it is what allows household wiring to work: every appliance receives full mains voltage regardless of how many other devices are plugged in.

Question 4

In the current divider formula for two parallel resistors, the current through R₁ is given by I₁ = I_total × R₁/(R₁ + R₂).

• T. True
• F. False

Answer: False

The correct formula is I₁ = I_total × R₂/(R₁ + R₂) — note that R₂ appears in the numerator, not R₁. This seems counterintuitive but follows directly from the physics: lower resistance means higher conductance means more current. The branch with smaller resistance carries the larger fraction of total current. The formula I₁ = I_total × R₁/(R₁+R₂) would give a larger share to the higher-resistance branch, which is the opposite of Ohm's law. Always check: the branch with smaller R should get the larger current fraction.

Question 5

Why is conductance G = 1/R the natural unit for analyzing parallel circuits, and what property of parallel circuits does it make immediately transparent?

Model answer: Conductance is natural for parallel circuits because conductances add directly: G_total = G₁ + G₂ + .... This mirrors how resistances add in series, revealing a symmetry between the two configurations. Using G makes the additive rule for parallel circuits as simple as the additive rule for series circuits. It also makes current distribution transparent: each branch carries current proportional to its conductance, so the highest-conductance (lowest-resistance) branch clearly carries the most current — which is exactly what we expect from 'current takes the path of least resistance.'

The series-parallel duality is the deeper insight: resistance is the additive quantity in series; conductance is the additive quantity in parallel. Switching between R and G notation just depends on which configuration you're analyzing. For parallel networks, conductance arithmetic is always simpler — total conductance is a sum, and each branch current I_i = V × G_i follows transparently from G_i's contribution to G_total.