Question

Show that for resistors connected in series, it is always the highest resistance that dissipates the most power, while for resistors connected in parallel, it is always the lowest resistance that dissipates the most power.

Short answer

Answer: In resistive circuits, the resistor with the highest resistance will dissipate the most power when connected in series, while the resistor with the lowest resistance will dissipate the most power when connected in parallel.

Step-by-step solution

Review the power formula for resistors

The power dissipated by a resistor can be calculated using the formula: \[P = I^2R \hspace{0.5cm} \textrm{or} \hspace{0.5cm} P = \frac{V^2}{R},\] where P is the power, I is the current passing through the resistor, V is the voltage across the resistor, and R is the resistance.

Analyze resistors in series

For resistors connected in series, the total resistance can be calculated using the formula, \[R_{tot} = R_1 + R_2 + ... + R_n,\] where \(R_{tot}\) is the total resistance and \(R_1, R_2, ..., R_n\) are the resistances of each individual resistor. Since the current is the same through all resistors in a series connection, we can write the power dissipated by each resistor as \[P_i = I^2 R_i.\] Since the current is constant, the power dissipated by a resistor will be directly proportional to its resistance. Therefore, the resistor with the highest resistance will dissipate the most power in a series connection.

Analyze resistors in parallel

For resistors connected in parallel, the total resistance can be calculated using the formula, \[\frac{1}{R_{tot}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n},\] where \(R_{tot}\) is the total resistance and \(R_1, R_2, ..., R_n\) are the resistances of each individual resistor. In a parallel connection, the voltage across each resistor is the same. Using the second power formula, we can write the power dissipated by each resistor as \[P_i = \frac{V^2}{R_i}.\] Since the voltage is constant, the power dissipated by a resistor will be inversely proportional to its resistance. Therefore, the resistor with the lowest resistance will dissipate the most power in a parallel connection.

Conclusion

We have shown that for resistors connected in series, the resistor with the highest resistance will dissipate the most power, and for resistors connected in parallel, the resistor with the lowest resistance will dissipate the most power. This is due to the direct proportionality between power and resistance for series connections, and the inverse proportionality between power and resistance for parallel connections.