Lesson 4 Flashcards — Series, Parallel, and Dividers

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1. What makes two elements in series? In parallel?

Series elements carry the same current — end to end with nothing branching between them. Parallel elements share the same two nodes, so they see the same voltage.

2. Give the equivalent resistance for resistors in series and in parallel.

Series: \(R_\text{EQ} = R_1 + R_2 + \cdots\). Parallel: \(1/R_\text{EQ} = 1/R_1 + 1/R_2 + \cdots\), or for exactly two, \(R_\text{EQ} = R_1R_2/(R_1+R_2)\).

3. How can you sanity-check a parallel combination?

The result is always smaller than the smallest branch. If it is not, you made an arithmetic error.

4. State the voltage divider rule.

\(V_k = V_s \cdot R_k / R_\text{EQ}\) — each resistor takes a share proportional to its own resistance. The biggest resistor gets the biggest voltage.

5. State the current divider rule.

The general rule works for any number of parallel branches: \(I_X = (R_\text{EQ,par}/R_X)\,I_\text{total}\), where \(R_\text{EQ,par}\) is the equivalent of the parallel group and the branch's own resistance sits in the denominator. Substituting \(R_\text{EQ,par} = R_1R_2/(R_1+R_2)\) gives the familiar two-branch form, \(I_1 = I_\text{total}\cdot R_2/(R_1+R_2)\), where the other resistor's value sits in the numerator because current prefers the easier path.

6. What is a classic Quiz 1 error with the current divider?

Reaching for the two-branch swap form when there are three or more branches — it is only valid for exactly two. With three or more, find \(R_\text{EQ,par}\) and use the general rule, keeping the branch's own resistance in the denominator. The two rules are mirror images: the voltage divider puts the element's own R on top, the general current divider puts it on the bottom.

7. Why does adding loads in parallel risk overloading a circuit?

Each added branch lowers \(R_\text{EQ}\), which raises the current drawn from the source. That is exactly how a circuit gets overloaded.

8. Why are the outlets in a building wired in parallel?

So every outlet sees the same 120 V (an AC value — defined properly in Lesson 6), and so unplugging one device does not interrupt the others. Old series Christmas lights show the alternative: one bulb fails and the whole string goes dark.

9. Describe the reduce-solve-expand method.

Label nodes, collapse series and parallel groups working back toward the source (redrawing each time), find \(I_s = V_s/R_\text{EQ}\), then expand outward with the divider rules to get the quantity asked for.

10. Why redraw the circuit after every reduction step?

Because it is far faster than finding a wrong answer later. Most reduction errors come from misreading which elements are still in parallel after a previous step.