Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Applications Involving Powers The power \(P\) dissipated in a resistance \(R\) through which is flowing a current \(I\) is \(P=I^{2} R .\) Therefore the power in a \(365-\Omega\) resistor carrying a current of \(0.5855 \mathrm{A}\) is \((0.5855)^{2}(365)\) W. Evaluate this power.

Short answer

The dissipated power in the resistor is approximately \(125.2\) W.

Step-by-step solution

Identify the Power Formula

The power dissipated in a resistance is given by the formula, where power is denoted as P, current as I, and resistance as R. In this case, use the following equation to calculate power: \(P=I^2R\).

Substitute the Values

Substitute the given values of current (I) and resistance (R) into the power formula. Current \(I=0.5855\) A and resistance \(R=365\) ohms.

Calculate the Power

Use the power formula with the substituted values to calculate the power: \(P=(0.5855)^2(365)\).

Perform the Calculation

First calculate the square of the current and then multiply by the resistance: \(P=(0.5855)^2(365) = (0.34280025)(365)\).

Retain Significant Digits

Since the current is given to four decimal places, and resistance is given to a whole number, the final calculated power should be reported with the same number of significant digits as the least precise measurement, which is three significant digits in this case: \(P \approx 125.2\) W.

Key concepts

Understanding Significant Digits

Significant digits, often referred to as significant figures, are crucial in expressing the precision of a measured or calculated quantity. These important figures provide insight into the reliability of the measurement and how much variation to expect in the last digit. When performing calculations, retaining the proper number of significant digits ensures that the precision of the result is consistent with the precision of the measurements used in the calculation.

For example, in the exercise, the current was given as 0.5855 A, with four decimal places, indicating high precision. On the other hand, the resistance was stated as a whole number (365 \(\Omega\)), suggesting it's precise to a lesser degree. According to the rules governing significant digits, when multiple quantities are multiplied or divided, the final answer must be rounded to the same number of significant digits as the factor with the fewest significant digits. In this case, the resistance's significant digits are the limiting factor, which means our result should be expressed with three significant digits, as it’s the whole number with no decimal places.

Electrical Resistance and Its Role

Electrical resistance is a measure of the opposition to the flow of electric current through a conductor. It is an intrinsic property of materials and is crucial in determining how effectively a given material can conduct electricity. Electrical resistance is measured in ohms (\(\Omega\)), named after the physicist Georg Simon Ohm.

In our example, the resistor has a resistance value of 365 \(\Omega\), which means it offers a significant level of opposition to the current flow. The resistance not only affects the current that flows for a given voltage (as described by Ohm's law) but also has a direct influence on the power dissipated by the resistor, which can be seen in the power dissipation formula, \(P=I^2R\). A higher resistance with a constant current will result in greater power dissipated as heat.

Ohm's Law Fundamentals

Ohm's law is a basic but fundamental principle in the field of electronics and electrical engineering. It states a direct proportionality between the voltage across a conductor and the current flowing through it, given that the physical conditions, like temperature, remain constant. The law is mathematically expressed as \(V = IR\), where \(V\) represents the voltage, \(I\) the current, and \(R\) the resistance. The simplicity of Ohm's law makes it a powerful tool in understanding and analyzing electrical circuits.

The relationship expressed by Ohm's law also underpins the power dissipation formula used in the exercise. When a voltage drives a current through a resistor, the energy is lost as heat, which is the dissipated power. By rearranging Ohm's law, one can derive the formula \(P=I^2R\), showing that power dissipation is proportionate to the square of the current and the resistance through which it flows. Therefore, small changes in current or resistance can lead to significant changes in the power dissipated, a factor that is particularly important in designing electrical and electronic systems, where managing heat is crucial.