# Circuitry problem solving

##### Intros

###### Lessons

- Introduction to Circuitry Problem Solving:
- A crash course review on electric circuits.
- Concept 1: How are individual resistances related to voltage drops?
- Concept 2: What can we conclude about resistors in series?
- Concept 3: What can we conclude about resistors in parallel?
- Concept 4: How can we determine the brightness of a lightbulb?

##### Examples

###### Lessons

**Problem Solving for Resistors and Voltage Drops**

The two circuits below are configured as shown. Circuit B includes an additional resistor R_{3}placed in parallel with R_{1}. Compare the values of V_{R1}and V_{R2}in circuit $B$ compared to circuit $A$ (no change, decrease, increase)?**Problem Solving for Lightbulb Brightness with Switches**

Two identical circuits are shown below, except Circuit A has an open switch and Circuit B has a closed switch. How does each lightbulb ($R_{1}, R_{2}, R_{3}, R_{4}$ and $R_{5}$) change in brightness compared to when the switch was open? (same, dimmer, or brighter)**Problem Solving for Complex Circuit with Missing Values****Solving for Lightbulbs and Terminal Voltage**

The circuit is connected to three identical lightbulbs:

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###### Topic Notes

In this lesson, we will learn:

- A brief review on voltage, current, and resistance
- Establishing 4 main concepts for problem solving:
- Concept #1: a smaller resistor uses up less voltage; a bigger resistor uses up more voltage
- Concept #2: the more resistors added in series with the battery into the circuit will increase the total equivalent resistance
- Concept #3: the more resistors added in parallel with the circuit will decrease the total equivalent resistance
- Concept #4: the brightness of a lightbulb is related to the voltage drop across it (as well as the power dissipated by it)
- Solving questions for more conceptual electric circuits questions:
- Using a combination of all previous concepts and formulas ($V, I, R,$ Ohm’s Law, $V_{term}$, Power)
- As well as applying the 4 main concepts

__Notes:__- Before facing
__problem solving__questions for electric circuits that are oftentimes just as**conceptual**as they are mathematical, one must have a firm understanding of the concepts of each lesson thus far: __Voltage:__staircase analogy, Kirchhoff’s Loop Rule, equal voltage in parallel__Current:__water analogy, Kirchhoff’s Junction Rule__Resistance:__calculating total resistance for series vs. parallel configurations- $R_{eq (series) = R_{1} + R_{2} + R_{3} + ... R_{n} = \sum_{k = 1}^n R_{k}}$
- $\frac{1}{R_{eq (parallel)}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . \frac{1}{R_{n}} = \sum_{k = 1}^n \frac{1}{R_{k}}$
- The conceptual relationships as defined by
__Ohm’s Law:__$V=IR$ - The shortcut for Ohm’s law; the
__voltage divider method__: $V_{x} = V_{total} \cdot \frac{R_{x}}{R_{total}}$ - The concept of
__terminal voltage__and calculations: $V_{term} = \epsilon - Ir$ __Power__: total power is additive, $P = \frac{E}{t}$ and $P = IV = I^{2}R = \frac{V^{2}}{R}$__Energy__: $E=P t$ and $E=I V t$- The 4 main concepts can be summarized as follows:
- I. The greater the resistance of a resistor, the more voltage that it uses up (and vice versa; a smaller resistor uses less voltage)
- II. The more resistors added in series, the greater the equivalent resistance
- III. The more resistors added in parallel, the lesser the equivalent resistance
- IV. The brightness of a lightbulb is related to the voltage it uses up (its voltage drop) as well as, the power dissipated by it
- The
__brightness__of a lightbulb is related to the amount of voltage that it uses up (voltage drop); the more voltage used, the brighter the light bulb - The voltage drop is dependent on current and resistance ($V=IR$)
- The
__brightness__of a lightbulb can also be understood as how hot the filament is burning - The incandescent lightbulb is transforming electrical energy into thermal and light energy; the rate of energy transformation is
__power__ - Power is dependent on voltage ($P=I V$) as well as current and resistance ($P=I^{2}R= \frac{V^{2}}{R}$ )
- When observing lightbulbs in
__series__: - Adding more lightbulbs in series will
**increase**the overall resistance, thus diminishing the total current—this leads to a smaller voltage drop across each lightbulb, causing aeffect__dimming__ - Opening a switch or having a single broken lightbulb in the series chain will cause all relevant lightbulbs turn off (the whole circuit will be compromised)
- When observing lightbulbs in
__parallel__: - Adding more lightbulbs in parallel will decrease the overall resistance, thus increasing the total current—the balance leads to relatively
brightness across all parallel lightbulbs**constant** - Opening a switch or having a single broken lightbulb will not compromise the whole circuit; only the relevant branch of the circuit will be affected (turned off)

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