Question

You are given two identical batteries and two pieces of wire, The red wire has a higher resistance than the black wire. You place the red wire across the terminals of one battery and the black wire across the terminals of the other battery, Which wire gets hotter?

Short answer

Answer: The black wire with lower resistance gets hotter when connected to identical batteries.

Step-by-step solution

Understand Ohm's law and Power dissipation

Ohm's law states that the voltage (V) across a resistor is equal to the current (I) flowing through it multiplied by its resistance (R): V = IR. Furthermore, the power (P) dissipated by a resistor can be calculated as P = IV, which can also be expressed as P = I^2R or P = V^2/R depending on the known variables.

Compare power dissipation in both wires

In this case, we have two identical batteries, which means they have the same voltage (V). Since the red wire has higher resistance (R1) than the black wire (R2), we can use P = V^2/R to understand the power dissipation in each wire. For the red wire: P1 = V^2 / R1 For the black wire: P2 = V^2 / R2 As R1 > R2, the value of P1 < P2. This means that the power dissipation in the black wire is higher than in the red wire.

Determining which wire gets hotter

Power dissipation in a resistor (or wire) can be interpreted as the conversion of electrical energy into thermal energy, resulting in heating. Therefore, the wire dissipating more power will get hotter. From Step 2, we know that the black wire (lower resistance) has higher power dissipation than the red wire. Thus, the black wire gets hotter when connected to the battery terminals.