Step-by-Step Guidance for Circuit Analysis and Resistance Calculation

Q2. Circuit Analysis and Resistance Calculation

Background

Topic: DC Circuits – Series and Parallel Resistors, Ohm’s Law, Power Dissipation, and Resistivity

This question tests your understanding of how to find equivalent resistance in a complex circuit, calculate current and power in specific resistors, and apply the resistivity formula to determine the resistance of a wire.

Key Terms and Formulas

• Ohm’s Law: $V = IR$

• Power Dissipated: $P = I^2R$ or $P = V^2/R$

• Resistors in Series: $R_{eq} = R_1 + R_2 + \ldots$

• Resistors in Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

• Resistivity Formula: $R = \rho \frac{L}{A}$, where $\rho$ is resistivity, $L$ is length, $A$ is cross-sectional area

Step-by-Step Guidance

1. Equivalent Resistance (a): - Identify which resistors are in series and which are in parallel. - Combine the parallel resistors first using $\frac{1}{R_{eq}} = \frac{1}{R_4} + \frac{1}{R_5} + \frac{1}{R_a}$, then add the series resistors.

2. Current through $R_5$ (b): - Once you have the equivalent resistance, use Ohm’s Law to find the total current from the battery: $I_{total} = \frac{V}{R_{eq}}$. - Use current division to find the current through $R_5$.

3. Power Dissipated by $R_5$ (c): - Use the current found in the previous step and the formula $P = I^2R$ to set up the calculation for power dissipated by $R_5$.

4. Resistance of Copper Wire (d): - Use the resistivity formula $R = \rho \frac{L}{A}$, where $\rho = 1.7 \times 10^{-8} \Omega\cdot m$, $L = 0.5$ m, and $A = \pi (\frac{d}{2})^2$ with $d = 1.6$ mm. - Convert all units to SI (meters, square meters).

Try solving on your own before revealing the answer!

Final Answers:

• a.) 10 $\Omega$

• b.) 0.125 A

• c.) 0.0625 W

• d.) 0.0042 $\Omega$

These results use the correct formulas for series/parallel resistors, Ohm's Law, power, and resistivity. The wire's resistance is much smaller than the other resistors, justifying the usual assumption to ignore it.