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

Question: Explain which resistor dissipates the most power in a series and parallel circuit, and justify your answer.

Step-by-step solution

Apply Ohm's Law for series resistors

The total resistance in a series circuit is obtained by simply adding all the individual resistances. Let's assume we have n resistors connected in series with resistances R1, R2, ..., Rn. The total resistance R_total = R1 + R2 + ... + Rn.

Calculate total current in the circuit

Using Ohm's Law (V = I * R), we can calculate the total current in the circuit. If V is the voltage across the circuit, the total current I = V / R_total.

Calculate power dissipation for each resistor

Power dissipation for a resistor can be calculated using P = I^2 * R. Since the current flowing through each resistor in a series circuit is the same, the resistor with the highest resistance will dissipate the most power. **For resistors connected in parallel:**

Apply Ohm's Law for parallel resistors

In a parallel circuit, the inverse of the total resistance is equal to the sum of the inverses of the individual resistances. Let's assume we have n resistors connected in parallel with resistances R1, R2, ..., Rn. The total resistance R_total = 1 / (1/R1 + 1/R2 + ... + 1/Rn).

Calculate voltage across each resistor

For parallel connected resistors, the voltage across each resistor is the same as the total applied voltage. If V is the voltage across the circuit, the voltage across each resistor is also V.

Calculate the current through each resistor

Using Ohm's Law (V = I * R), we can find the current through each resistor. For the ith resistor, the current I_i = V / R_i.

Calculate power dissipation for each resistor

Power dissipation for a resistor can be calculated using P = I^2 * R. Since the voltage across each resistor in a parallel circuit is the same, the resistor with the lowest resistance will have the highest current and thus, dissipate the most power. By following these steps, we have shown that in a series circuit, the resistor with the highest resistance dissipates the most power, and in a parallel circuit, the resistor with the lowest resistance dissipates the most power.