Question

Manipulating Functions.The power \(P\) dissipated in a resistor is given by \(P=I^{2} R .\) Write \(R=f(P, I)\).

Short answer

R = P / I^2

Step-by-step solution

Understand the given equation

Given the power dissipation equation for a resistor, which is P = I^2 * R, where P is the power, I is the current, and R is the resistance.

Isolate the variable R

The task is to express R as a function of P and I. To do this, divide both sides of the equation by I^2 to isolate R on one side of the equation.

Solve for R

After dividing both sides of the equation by I^2, the equation becomes R = P / I^2.

Key concepts

Power Dissipation in Resistors

Understanding how resistors use energy in an electrical circuit is foundational to grasping various principles in electronics and physics. Power dissipation in resistors is a measure of the energy converted into heat as current flows through a resistor. This concept is crucial in designing circuits to prevent damage from overheating.

The formula that describes the power dissipation in a resistor is given by the expression \(P = I^2R\), where \(P\) stands for power measured in watts (W), \(I\) is the current in amperes (A), and \(R\) is the resistance in ohms (Ω). The equation reveals that power increases with the square of the current, meaning if you double the current, the power dissipation will quadruple. This also implies a directly proportional relationship between power and resistance; as resistance increases, power does likewise.

Key Takeaway

It's critical for engineers to consider this relationship when designing circuits, as components have limits on the power they can handle without sustaining damage. Thermal management strategies, such as heat sinks or cooling systems, might be necessary when a resistor must endure high levels of power dissipation.

Isolating Variables

When working with equations, one often needs to isolate a variable to solve for it—this means getting the variable on one side of the equation by itself. Isolating variables is a fundamental technique in algebra that enables us to solve equations and understand relationships between different quantities.

In the context of the power dissipation equation, \(P = I^2R\), our task is to isolate resistance \(R\). To achieve this, we manipulate the equation so that \(R\) appears alone on one side. We do this by dividing both sides by the square of the current \(I^2\), yielding \(R = \frac{P}{I^2}\). This operation does not change the equality because we are performing the same action on both sides of the equation.

Understanding Operations

These operations are reversible and maintain the balance of the equation, reflecting the fundamental principle of algebra that what you do to one side, you must do to the other. Crafting such an understanding is pivotal when attempting to restructure equations to solve for a particular variable.

Solving Equations

Solving equations is the process of finding the values for the variables that make the equation true. For most simple equations, this involves a sequence of steps like adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression.

In the given exercise, solving for resistance involves just one step: dividing both sides by the square of the current. Remember, solving equations often requires looking at the equation to determine the best course of action to isolate the desired variable. In our case, since the power \(P\) is equal to the product of the current squared \(I^2\) and the resistance \(R\), dividing by \(I^2\) makes sense. The solution is then \(R = \frac{P}{I^2}\).

Verifying Solutions

It's important to verify the solution by plugging it back into the original equation to see if it satisfies the equation. In printed circuit board (PCB) design and other practical applications, these calculations help ensure safe working conditions and protect components from damage by verifying that the values are within acceptable ranges.