Circuit Lab 3: Resistance and Current in Series Circuits

Measurement of Resistive Components: The primary objective is to measure the resistance of individual resistors using a digital multimeter to establish baseline values for the experiment.
Investigation of Series Resistance: To experimentally observe and document how the total resistance ( $R_{total}$ ) of a circuit changes as additional resistors are introduced in a series configuration.
Analysis of the Current-Resistance Relationship: To examine the inverse relationship between the total resistance of the circuit and the resulting electrical current ( $I$ ) flowing through it.
Derivation of Fundamental Laws: To use the collected experimental data to derive Ohm's Law ( $V = I \times R$ ).
Verification of Series Circuit Theory: To confirm the theoretical principle that the total resistance in a series circuit is equal to the algebraic sum of the individual resistances of the components involved ( $R_{total} = R_1 + R_2 + ... + R_n$ ).

Battery Eliminator: A power source set to an output of $3V$ .
Multimeter: A digital measuring device used to quantify resistance in Ohms ( $\Omega$ ), potential difference in Volts ( $V$ ), and current in Amperes ( $A$ ).
Circuit Board: The physical platform used to assemble the electrical components.
Switch: A single (1) component used to open or close the circuit to control current flow.
Connecting Wires: Used to facilitate electrical connections between components.
Resistors (Approximate Values): - Green Resistor: Approximately $5\,\Omega$ - Blue Resistor: Approximately $10\,\Omega$ - Red Resistor: Approximately $200\,\Omega$ (Note: The datasheet likely refers to nominal values, while experimental data shows values closer to $20\,\Omega$ for the Red resistor).

Before building the circuits, the multimeter was used to determine the precise resistance of each provided component: - Green Resistor: $5.1\,\Omega$ - Blue Resistor: $10.1\,\Omega$ - Red Resistor: $20.2\,\Omega$

Circuit Assembly Guidelines: - Circuits must be built using the battery eliminator, a switch, and specific resistor combinations in a series arrangement. - For each specific configuration, the switch is opened to measure total current. - The multimeter is used to measure the total voltage across the entire circuit. - Individual voltage drops across each specific resistor in the series are measured. - The total resistance of the configuration is measured directly using the multimeter.
Resistor Configurations (Circuits 1-7): - Circuit 1: Contains 1 Green resistor. - Circuit 2: Contains 1 Blue resistor. - Circuit 3: Contains 1 Red resistor. - Circuit 4: Contains 1 Green resistor and 1 Blue resistor in series. - Circuit 5: Contains 1 Green resistor and 1 Red resistor in series. - Circuit 6: Contains 1 Blue resistor and 1 Red resistor in series. - Circuit 7: Contains 1 Blue resistor, 1 Green resistor, and 1 Red resistor in series.

Circuit	Total Voltage ( $V$ )	Voltage Green ( $V$ )	Voltage Blue ( $V$ )	Voltage Red ( $V$ )	Total Current ( $A$ )	Total Resistance ( $\Omega$ )
1	$3.0$	$3.0$	na	na	$0.588$	$5.1$
2	$3.0$	na	$3.0$	na	$0.297$	$10.1$
3	$3.0$	na	na	$3.0$	$0.149$	$20.2$
4	$3.0$	$1.00$	$1.99$	na	$0.197$	$15.2$
5	$3.0$	$0.61$	na	$2.40$	$0.119$	$25.3$
6	$3.0$	na	$1.00$	$2.00$	$0.099$	$30.3$
7	$3.0$	$0.43$	$0.86$	$1.72$	$0.085$	$35.4$

Key Observations from Data: - In every circuit configuration, the Total Voltage remained constant at $3.0\,V$ . - In series circuits with multiple resistors, the sum of the individual voltage drops across the resistors equals the total voltage (e.g., Circuit 7: $0.43 + 0.86 + 1.72 = 3.01\,V$ , approximately equal to the source voltage). - The Total Resistance of a series circuit is the sum of the individual resistors (e.g., Circuit 4: $5.1\,\Omega + 10.1\,\Omega = 15.2\,\Omega$ ). - As the total resistance in the circuit increases, the total current flowing through the circuit decreases, indicating an inverse relationship.

Graph 1: Current vs. Total Resistance: - X-axis: Total Resistance ( $\Omega$ ). - Y-axis: Current ( $A$ ). - Analysis: A best-fit curve should be drawn through the data points. This typically results in a hyperbola, demonstrating that current is inversely proportional to resistance when voltage is constant ( $I \propto \frac{1}{R}$ ).
Graph 2: Current vs. Reciprocal of Resistance: - X-axis: $1 / \text{Total Resistance}$ ( $1/\Omega$ ). - Y-axis: Current ( $A$ ). - Analysis: Plotting current against the inverse of resistance should result in a linear relationship. The slope of this line corresponds to the constant voltage of the system ( $V$ ), as per the rearranged Ohm's Law: $I = V \times (\frac{1}{R})$ .