Problem 20 - Technical Mathematics with Calculus

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 current through a resistor is increased by \(28 \%,\) by what percent will the power increase?

Step-by-Step Solution

Answer

The power will increase by approximately 63.84%.

Step 1: Understanding the Problem

We are given that power dissipation in a resistor is directly proportional to the square of the current through it, which is expressed by the equation, \(P = I^2 R\). We need to find the percentage increase in power when there is a 28% increase in current.

Step 2: Calculating the Final Current

First, calculate the final current after an increase of 28%. If the initial current is \(I\), the final current \(I'\) after the increase will be \(I' = I(1 + 0.28) = 1.28I\).

Step 3: Calculating the Final Power

Now, we can find the final power \(P'\) using the final current \(I'\). Using the given formula \(P = I^2 R\), the final power \(P'\) will be \(P' = (I')^2 R = (1.28I)^2 R = 1.6384I^2 R\). Since \(P = I^2 R\), replacing we get \(P' = 1.6384P\).

Step 4: Finding the Percentage Increase in Power

The percentage increase in power is given by the ratio of the increase in power to the original power times 100%. This can be calculated as follows: Percentage increase in power = \(\left(\frac{P' - P}{P}\right) \times 100% = \left(\frac{1.6384P - P}{P}\right) \times 100% = (1.6384 - 1) \times 100% = 63.84%\).

Key Concepts

Understanding Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity that relates the current through a conductor to the voltage across it and the resistance of the conductor. It can be expressed with the formula
\( V = IR \)
where
\( V \) represents voltage in volts,
\( I \) is the current in amperes, and
\( R \) signifies resistance in ohms. This law indicates that the current flowing through a circuit is directly proportional to the voltage and inversely proportional to the resistance.
Understanding Ohm's Law is crucial because it underpins much of what we know about circuits and how they function. For instance, when we calculate the power dissipated by a resistor, we often rearrange Ohm's Law to express it in terms of power. This understanding enables us to predict how changes in a circuit, such as in the current or resistance, will affect the power dissipation. Let's explore how this works in the context of our exercise, where the power is related to the square of the current.

Performing Percentage Increase Calculation

Percentage increase calculations are an important part of understanding changes in quantities, especially in fields like physics, economics, and everyday scenarios. The percentage increase is found by comparing the change in a value to the original value and then expressing this rise as a part of 100.
To calculate the percentage increase, you can use the following formula:
\( \text{Percentage Increase} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\% \)
To apply this to our problem, the original value is the power dissipated by the resistor before the current increased, and the new value is the power after the current has increased. The step-by-step solution above demonstrates this calculation, giving us a percentage increase in power. Understanding how to perform percentage increase calculations is essential for interpreting results and making comparisons, such as deciding if an increase in power dissipation is significant or evaluating the efficiency of an electrical device after a change in its operation.

Exploring the Power-Current Relationship

In electrical circuits, there is a specific relationship between power and current that is very relevant to the problem discussed. The power dissipated by an electrical component, like a resistor, is fundamentally tied to the current flowing through it. This relationship is often expressed by the equation:
\( P = I^2 R \)
where
\( P \) is power in watts,
\( I \) is current in amperes, and
\( R \) is resistance in ohms. As shown in this formula, power is proportional to the square of the current times the resistance.
Understanding the power-current relationship is important because it indicates how changes in current affect power dissipation. For instance, a small increase in current can cause a significant increase in power dissipation due to the current being squared in the relationship. This is particularly important in the design and operation of electrical systems to ensure components do not overheat and function within their capacities. In the context of our exercise, when the current is increased by 28%, this relationship allows us to calculate how much the power dissipation will increase, which turns out to be a 63.84% increase—a much larger percentage than the increase in current itself, due to the squared term in the power equation.