Resistors and Resistor Combinations – Part 1

Hook: Opening Question

  • Setup: Two identical bulbs (A & B) wired in the configuration shown in Hewitt Fig. 1 (parallel branches, switch controlling branch B).

  • Prompt: “What happens to the brightness of bulb A when the switch is closed and bulb B lights up?”

  • Key insight previewed: In parallel, voltage across each branch remains identical to battery’s emf; therefore current through A is unaffected by closing the switch to B.

Overview of Circuit Connections

  • Only two fundamental ways to wire resistive elements:

    • Series connection – single continuous path for charge flow.

    • Parallel connection – multiple branches that share common terminals; current divides.

  • Practical ubiquity: Every electronic device (phones, computers, household wiring) relies on deliberate series & parallel combinations to control current, voltage drops, and power distribution.

  • Visual summary supplied by Giancoli Fig. 2.

Series Connection

  • Characteristics

    • Single path ⇒ same current everywhere: I = I1 = I2 = I_3 = \ldots (eqn 1)

    • Voltage divides: V = V1 + V2 + V_3 + \ldots (eqn 2)

    • Equivalent resistance adds: R{eq} = R1 + R2 + R3 + \ldots (eqn 3)

  • Derivation sketch (Giancoli Fig. 3)

    1. Ohm’s law on each resistor: Vi = Ii R_i

    2. Energy conservation: total battery voltage equals sum of individual drops.

    3. Since current is common, factor it out to obtain additive rule for R_{eq}.

  • Consequences

    • Adding more resistors always increases R_{eq}.

    • Current through entire loop decreases for a fixed supply when more series elements are added.

  • Everyday analogy: Water flowing through one narrow pipe after another—each pipe adds to total resistance.

Parallel Connection

  • Characteristics (Giancoli Fig. 4)

    • Voltage is common across all branches: V = V1 = V2 = V_3 = \ldots (eqn 5)

    • Current splits at junctions: I = I1 + I2 + I_3 + \ldots (eqn 4)

    • Equivalent conductances add leading to:
      \frac{1}{R{eq}} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + \ldots (eqn 6)

  • Derivation sketch

    1. From charge conservation: incoming current equals sum of outgoing branch currents.

    2. Apply Ohm’s law in each branch using common voltage.

    3. Rearrange to express 1/R_{eq} as sum of reciprocals.

  • Practical note: A single open branch does not break the rest of the circuit—major reason house/appliance wiring is done in parallel.

  • Resistance intuition: Adding more parallel paths decreases overall R_{eq} (easier for charges to flow).

Application Back to Hook Question

  • Closing switch introduces bulb B in parallel with bulb A.

  • Because VA = VB = V_{battery}, current through A stays the same → brightness unchanged.

  • Equivalent resistance of the circuit halves (two identical resistances in parallel):
    R_{eq,\text{new}} = \frac{R}{2}

  • Battery now supplies double the current, but transistor splits; each branch still gets original current I = V/R.

Conceptual Examples & Their Lessons

Example 1 – Are the three diagrams in Hewitt Fig. 5 equivalent?

  • Yes. All depict three separate conductive paths tied directly across the battery → pure parallel wiring.

  • Take-away: Visual variations (wire routing) can mask underlying equivalence; always trace terminals.

Example 2 – Ammeter Placements in Parallel Network (Hewitt Fig. 6)

  • Identical bulbs ⇒ identical branch resistances.

  • With common voltage each draws same current I_b.

  • Ammeter readings:

    • At position A (before split) ⇒ I = 3I_b

    • At B (in single branch) ⇒ I_b

    • At C (another branch) ⇒ I_b

  • Ranking greatest → least: A = B = C?

    • Text states A = B = C because the graphic likely shows only two branches; for three identical, one obtains different totals. Accept lesson: currents in equal-resistance parallel branches are equal.

Example 3 – Voltmeter Across Bulbs in Series (Hewitt Fig. 7)

  • All bulbs identical; total battery voltage V_{bat} splits equally.

  • Diagram A: single bulb ⇒ voltmeter reads V_{bat}.

  • Diagram B: two bulbs ⇒ each drops V_{bat}/2.

  • Diagram C: three bulbs ⇒ each drops V_{bat}/3.

  • Ranking voltmeter readings: A > B > C.

  • Moral: In series, increasing the count of identical loads reduces individual voltage drops proportionally.

Connections to Broader Principles

  • Charge conservation underlies series-current equality and junction-current splitting.

  • Energy conservation governs voltage summation in series loops.

  • Ohm’s law V = IR bridges microscopic material resistance with macroscopic circuit behavior.

  • Everyday design:

    • Series: fuse placement, current-limiting resistors.

    • Parallel: domestic lighting, computer motherboard power rails—failure of one branch does not kill entire system.

  • Ethical/Practical implication: Proper understanding prevents hazards such as overloading (too many low-resistance parallel devices drawing excessive total current).

Numerical/Formula Summary

  • Series:

    • Current: I = \text{constant}

    • Voltage: V = \sum V_i

    • Resistance: R{eq} = \sum Ri

  • Parallel:

    • Voltage: V = \text{constant}

    • Current: I = \sum I_i

    • Resistance: \frac{1}{R{eq}} = \sum \frac{1}{Ri}

Study Tips

  • When faced with a complex network, isolate junction pairs (common terminals); everything sharing those terminals is in parallel.

  • Collapse step-by-step: reduce simple series/parallel chunks into single R, redraw, repeat.

  • Keep units: volts (V), amperes (A), ohms (Ω).

  • Memorize extreme cases:

    • Two equal resistors in parallel ⇒ R_{eq} = R/2.

    • Two equal resistors in series ⇒ R_{eq} = 2R.

References

  • Hewitt, P. G. Conceptual Physics, 12th ed., 2015.

  • Giancoli, D. C. Physics: Principles with Applications, 7th ed., 2013.

  • Cutnell, J. D., & Johnson, K. W. Physics, 9th ed., 2012.