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 gets hotter than the red wire when connected across the terminals of identical batteries.

Step-by-step solution

Identify the relevant formulas

We'll use Ohm's Law and the Power equation as follows: - Ohm's Law: V = I * R, where V is voltage, I is current, and R is resistance - Power equation: P = I^2 * R, where P is power, I is current, and R is resistance

Assign known values

The exercise informs us that both batteries are identical. Let's assume that each battery has a voltage, V. The red wire has a higher resistance (R_red) than the black wire (R_black).

Calculate current through each wire

Using Ohm's Law, we can calculate the current through each wire: I_red = V / R_red I_black = V / R_black Since R_red > R_black, it follows that I_red < I_black.

Calculate power dissipation for each wire

Now we will use the Power equation to calculate the power dissipation (which is related to heat production) in each wire: P_red = (I_red)^2 * R_red P_black = (I_black)^2 * R_black

Compare power dissipation

To determine which wire gets hotter, we will compare the power dissipation in each wire. Notice that P_red = ((V / R_red)^2) * R_red = V^2 / R_red Similarly, P_black = ((V / R_black)^2) * R_black = V^2 / R_black Since R_red > R_black, it follows that P_red < P_black.

Draw conclusion

Based on the calculations, the power dissipation in the black wire (P_black) is greater than the power dissipation in the red wire (P_red). More power dissipation means more heat production. Therefore, the black wire gets hotter than the red wire when connected across the terminals of the identical batteries.

Key concepts

Ohm's Law

When studying electrical circuits, understanding Ohm's Law is essential. It’s a fundamental principle that describes the relationship between voltage, current, and resistance. According to Ohm's Law, which is expressed as the equation \( V = I \times R \), the voltage (\( V \)) across a conductor is directly proportional to the current (\( I \)) flowing through it, and this relationship is governed by the resistance (\( R \)) of the conductor.

In simple terms, if you know two of these values, you can always solve for the third. Ohm's Law becomes incredibly useful when trying to determine why one wire may get hotter than another, as in our textbook exercise. The higher the resistance in the red wire, the lower the current it allows for a given voltage, compared to the black wire with lower resistance.

Power dissipation

Power dissipation in electrical conductors is a critical concept, particularly in understanding heat generation. Power, measured in watts (\( W \)), can be calculated by the equation \( P = I^2 \times R \), where \( P \) represents the power dissipated, \( I \) is the current, and \( R \) is the resistance.

What does this mean for our wires? Since power equates to energy usage per unit of time, and here it’s directly tied to the generation of heat, the wire dissipating more power will get hotter. The amount of current flowing through the wire plays a significant role due to it being squared in the power formula, meaning that even small increases in current can lead to substantial increases in power dissipation and therefore more heat.

Current and resistance relationship

The interconnection between current and resistance is part of the heart of Ohm’s Law. As resistance increases, the current decreases for a given voltage, which can be showcased by the formulas obtained from Ohm's Law: \( I_{red} = \frac{V}{R_{red}} \) and \( I_{black} = \frac{V}{R_{black}} \).

Considering a constant voltage, the red wire's higher resistance leads to a smaller current flowing through it than the black wire, which has a lower resistance. This relationship is vital to predict how electrical components behave in a circuit, and it is a key factor in understanding why certain materials and specifications are chosen for specific electrical applications.

Heat generation in conductors

Heat generation in conductors is an outcome of electrical resistance in materials. As electric current passes through a conductor, the interaction between the moving electrons and the fixed particles in the conductor can cause collisions, leading to energy being released as heat, a process described by Joule's law.

The textbook exercise demonstrates this concept as we compare the heat produced by two wires of different resistances. A crucial point to consider is that even though the red wire has greater resistance, it does not necessarily generate more heat than the black wire. The power dissipation formula \( P = I^2 \times R \) shows us that it's not just the resistance, but also the square of the current that determines the heat produced. Generally, a wire with higher current will generate more heat, assuming other factors remain constant.