Question

From Eq. \(1066, P=I^{2} R,\) we see that power \(P\) dissipated in a resistor varies directly as the square of the current \(I\) in the resistor. If the power dissipated in a resistor is \(486 \mathrm{W}\) when the current is \(2.75 \mathrm{A}\), find the power when the current is \(3.45 \mathrm{A}\)

Short answer

The power dissipated when the current is 3.45 A is approximately 648 W.

Step-by-step solution

Identify the Given Information

Note the given information: power dissipated at 2.75 A is 486 W, and the formula relating power and current is given by the equation \(P=I^{2}R\). We want to find the power when the current is 3.45 A.

Calculate the Resistance Using the Given Power and Current

Use the power formula to find the resistance. With \(P = 486 W\) and \(I = 2.75 A\), the resistance \(R\) can be found with \(R = \frac{P}{I^2}\).

Plug in the Known Values to Find the Resistance

To find \(R\), plug the known values into the resistance formula to get \(R = \frac{486 W}{(2.75 A)^2}\). Calculate the resistance.

Calculate the New Power Using the Current of 3.45 A

Now that we have the resistance, use the original power formula \(P=I^2R\) with \(I = 3.45 A\) to find the new power dissipated.

Substitute the Found Resistance into the Power Formula

Insert the calculated value of \(R\) from step 3 into the formula \(P=I^2R\) and calculate the power at the new current of 3.45 A.

Key concepts

Ohm's Law

Understanding the relationship between current, voltage, and resistance is foundational to comprehending electrical circuits. Ohm's Law makes this relationship clear and is stated as \( V = IR \), where \( V \) represents voltage in volts, \( I \) is the current in amperes, and \( R \) is the resistance in ohms. Essentially, the law posits that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. This principle is crucial not only for predicting the behavior of circuits but also for understanding power dissipation in resistors.

For example, if you increase the voltage keeping the resistance constant, the current will rise proportionally, according to Ohm's Law. As the current increases, you would expect more power to be dissipated in the form of heat within the resistor. To tie this to our original exercise, once we know the resistance, we can manipulate the current to find variations in power dissipated, all thanks to Ohm's Law.

Power Dissipation in Resistors

When a current flows through a resistor, electrical energy is converted into heat, a process we call power dissipation. This power, denoted as \( P \), is measured in watts (W). The power dissipated by a resistor can be calculated using Ohm's Law and the formula \( P = I^2R \), where \( I \) is the current in amperes and \( R \) is the resistance in ohms. The \( I^2 \) term highlights that power increases with the square of the current. This squared relationship means that a small increase in current can cause a large increase in power dissipation, which is essential to understand for electrical safety and efficiency.

In the context of our original problem, identifying the resistor's resistance allows us to determine how power dissipation changes with different currents. It's a direct variation — as current increases, so does the power dissipated, essential for the accurate design and management of electrical circuits.

Direct Variation

Direct variation is a fundamental concept in algebra that refers to a proportional relationship between two variables: when one variable changes, the other changes by a constant multiple. In other words, the ratio of one variable to the other remains constant. Mathematically, this is represented as \( y = kx \), where \( y \) and \( x \) are the variables, and \( k \) is the constant of variation or the proportionality constant.

Connecting this to electrical power calculation, we have seen in the exercise that the power dissipated in a resistor (\( P \)) varies directly as the square of the current (\( I \)). This direct variation is evident in the formula \( P = I^2R \), where \( R \) remains a constant. So as the current increases or decreases, the power dissipation (\