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Chapter 3: Problem 8

dissipated, \(P(V)\), by a resistor is given by $$ P(V)=\frac{V^{2}}{R} $$ Find \(P(I R)\).

Short Answer

Expert verified
P(I R) = I^{2} R

Step by step solution

01

Identify given formula

The power dissipated by a resistor is given by the formula: \[ P(V) = \frac{V^{2}}{R} \]
02

Substitute the expression for voltage

The voltage across the resistor can be expressed as the product of current (I) and resistance (R). Therefore, substitute \(V = I R\) into the power formula: \[ P(I R) = \frac{(I R)^{2}}{R} \]
03

Simplify the expression

Simplify the expression by performing the squaring operation: \[ P(I R) = \frac{I^{2} R^{2}}{R} \]
04

Divide by resistance

Cancel out one resistance term (R) in the numerator and denominator: \[ P(I R) = I^{2} R \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ohm's Law
Ohm's Law is an essential principle in electrical engineering that describes the relationship between voltage, current, and resistance in an electrical circuit. The law is formulated as:
\(V = I R\).
This formula tells us that:
  • V (voltage) is the potential difference across the resistor.
  • I (current) is the flow of electric charge through the resistor.
  • R (resistance) opposes the flow of current.
By understanding this relationship, we can derive other equations and solve various problems in electrical circuits. For instance, if we know the current flowing through a resistor and the resistance, we can easily calculate the voltage across it.
Electrical Power
Electrical Power in a resistor explains how much energy is used or converted into heat in a circuit. The basic formula to find power is:
\(P = IV\).
However, we can use Ohm's Law to derive other useful forms of this equation. Given \( V = I R \), we can substitute this into the power formula:
\(P = I (IR)\)
Further simplifying, we get:
\(P = I^{2} R\).
This form is particularly helpful when we only know the current and the resistance. Similarly, if we only know the voltage and resistance, we can use the formula seen in the exercise:
\(P = \frac{V^{2}}{R}\).
  • High current results in higher power dissipation.
  • Higher resistance at a constant voltage or current also leads to higher power dissipation.
Understanding these relationships helps in designing circuits that operate efficiently without overheating.
Resistor Formulas
Various formulas revolve around resistors in electrical circuits, helping us to analyze and understand their behavior under different conditions.
Some key formulas include:
  • Ohm's Law: \( V = I R \)
  • Power Dissipation (using voltage): \( P = \frac{V^{2}}{R} \)
  • Power Dissipation (using current): \( P = I^{2} R \)
When solving problems involving resistors, knowing these formulas allows us to convert and substitute various quantities to find unknown values. For example, in the exercise, we started with the power formula using voltage, \( P = \frac{V^{2}}{R} \), and substituted the voltage \( V = I R \) to find the power using current:
\( P = \frac{(IR)^{2}}{R} \).
By simplifying, we derived:
\( P = I^{2}R \).
This highlights how interconnected these formulas are and how they can be adapted based on the given information.

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