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: $I = I<em>1 = I</em>2 = I_3 = \ldots$ (eqn 1)

  • Voltage divides: $V = V<em>1 + V</em>2 + V_3 + \ldots$ (eqn 2)

  • Equivalent resistance adds: $R<em>{eq} = R</em>1 + R<em>2 + R</em>3 + \ldots$ (eqn 3)

• Derivation sketch (Giancoli Fig. 3)

  1. Ohm’s law on each resistor: $V<em>i = I</em>i R_i$

  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: $V = V<em>1 = V</em>2 = V_3 = \ldots$ (eqn 5)

  • Current splits at junctions: $I = I<em>1 + I</em>2 + I_3 + \ldots$ (eqn 4)

  • Equivalent conductances add leading to:
    $\frac{1}{R<em>{eq}} = \frac{1}{R</em>1} + \frac{1}{R<em>2} + \frac{1}{R</em>3} + \ldots$ (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 $V<em>A = V</em>B = V_{battery}$, 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: $R<em>{eq} = \sum R</em>i$

• Parallel:

  • Voltage: $V = \text{constant}$

  • Current: $I = \sum I_i$

  • Resistance: $\frac{1}{R<em>{eq}} = \sum \frac{1}{R</em>i}$

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.