Power Rating of a Resistor. The power rating of a resistor is the maximum power the resistor can safely dissipate without too great a rise in temperature and hence damage to the resistor. (c) A 100.0-Ω and a 150.0-Ω resistor, both rated at 2.00 W, are connected in series across a variable potential difference. What is the greatest this potential difference can be without overheating either resistor, and what is the rate of heat generated in each resistor under these conditions?

Verified step by step guidance

1. First, understand that when resistors are connected in series, the same current flows through each resistor. The power dissipated by a resistor is given by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><msub><mi>I</mi><mi>i</mi></msub><mo>^</mo><mn>2</mn><msub><mi>R</mi><mi>i</mi></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>i</mi></msub></math> is the power, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>i</mi></msub></math> is the current, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>i</mi></msub></math> is the resistance.
2. Since both resistors are rated at 2.00 W, the maximum current through each resistor can be calculated using the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>i</mi></msub><mo>=</mo><msqrt><mfrac><msub><mi>P</mi><mi>i</mi></msub><msub><mi>R</mi><mi>i</mi></msub></mfrac></msqrt></math>. Calculate the maximum current for each resistor using their respective resistances.
3. Once the maximum current is determined, use the smallest current value to ensure neither resistor exceeds its power rating. This is because the same current flows through both resistors in series.
4. Calculate the total resistance of the series circuit using the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math> are the resistances of the individual resistors.
5. Finally, calculate the maximum potential difference across the series circuit using Ohm's Law: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>I</mi><mi>R</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> is the potential difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> is the current, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is the total resistance. This will give you the greatest potential difference that can be applied without overheating either resistor.

Key Concepts

Power Dissipation in Resistors

Power dissipation in a resistor is the process by which electrical energy is converted into heat energy. It is calculated using the formula P = I^2R, where P is the power, I is the current, and R is the resistance. Understanding this concept is crucial for determining the maximum power a resistor can handle without overheating.

Series Circuit Characteristics

In a series circuit, resistors are connected end-to-end, and the same current flows through each resistor. The total resistance is the sum of individual resistances, and the voltage across the circuit is divided among the resistors. This concept helps in calculating the potential difference across each resistor and ensuring none exceeds its power rating.

Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, expressed as V = IR. This fundamental principle is essential for calculating the current in the circuit and subsequently the power dissipated by each resistor, ensuring they operate within their rated power limits.