Problem 62 - Fundamentals of Physics

Question

A resistor with a potential difference of \(200 \mathrm{~V}\) across it transfers electrical energy to thermal energy at the rate of \(3000 \mathrm{~W}\). What is the resistance of the resistor?

Step-by-Step Solution

Answer

The resistance of the resistor is approximately \( 13.33 \Omega \).

Step 1: Understand the Given Information

We have a potential difference (voltage, \( V \)) of \( 200 \mathrm{~V} \) across a resistor. The power, \( P \), which is the rate of energy conversion, is given as \( 3000 \mathrm{~W} \). We need to find the resistance, \( R \), of the resistor.

Step 2: Recall the Power Formula

Remember that electrical power (\( P \)) in terms of voltage (\( V \)) and resistance (\( R \)) is given by the equation: \[ P = \frac{V^2}{R} \] Where \( P \) is power in watts, \( V \) is voltage in volts, and \( R \) is resistance in ohms.

Step 3: Rearrange the Formula to Find Resistance

We need to solve for resistance \( R \). Rearranging the power formula gives us: \[ R = \frac{V^2}{P} \] This formula will allow us to compute the resistance using the given power and voltage values.

Step 4: Substitute the Values and Solve

Now substitute the given values into the rearranged formula:\[ R = \frac{200^2}{3000} \] First calculate \( 200^2 \), which is \( 40000 \). Then compute the value of \( R \):\[ R = \frac{40000}{3000} \approx 13.33 \] So, the resistance is approximately \( 13.33 \Omega \).

Key Concepts

Resistance Calculation

When determining resistance, one can turn to Ohm's Law and the power formula to uncover this essential electrical property. Within circuits, finding resistance is pivotal as it quantifies how strongly a resistor opposes the flow of current.
Ohm's Law offers the fundamental relationship:

• Voltage (\( V \))
• Current (\( I \))
• Resistance (\( R \))

However, in the given problem, where electrical power (\( P \)) is known, we employ the power formula: \[ P = \frac{V^2}{R} \] This equation shows the interplay between voltage, power, and resistance.
To find resistance (\( R \)), we rearrange the formula, resulting in: \[ R = \frac{V^2}{P} \] Using this formula, one measures how much the resistor reduces the current for a specific voltage. With voltage (\( 200 \mathrm{~V} \)) and power (\( 3000 \mathrm{~W} \)), our calculation yields a resistance of approximately \( 13.33 \Omega \). Understanding this concept is vital as resistance influences circuit performance and stability significantly.

Electrical Power

Electrical power (\( P \)) plays an essential role in electrical systems. It represents the rate at which energy is transferred or used within a circuit. It effectively measures how much electrical energy is being converted into another form, like thermal energy, as seen in resistors.
Power can be expressed in terms of voltage and resistance by the formula: \[ P = \frac{V^2}{R} \] This equation shows that power increases with voltage squared, meaning higher voltages will significantly impact the power output if resistance remains constant.
In practical applications, electrical power drives devices and circuits, from simple light bulbs to complex machines. Knowing how to calculate power from given circuit parameters like voltage and resistance enables effective design and safety measures to be implemented within devices.

• This ensures they perform optimally.
• Prevents potential hazards from excess power dissipation.

Understanding power calculations helps ensure the energy efficiency and longevity of electrical components.

Voltage-Current Relationship

The relationship between voltage and current is a central tenet of Ohm's Law. In the world of electricity, voltage describes the potential difference that causes current to flow.
Current, on the other hand, is the actual flow of electric charges. Ohm's Law binds them together with resistance through the equation: \[ V = IR \] Here, the voltage (\( V \)) across a resistor is equal to the product of the current (\( I \)) through it and its resistance (\( R \)).

• This means for a constant resistance, as voltage increases, current also increases.
• Conversely, for a given voltage, higher resistance will reduce the current.

Understanding the voltage-current relationship is crucial for tasks such as determining how much current flows in a circuit when a particular voltage is applied, or adjusting the voltage to achieve a desired current. These insights are key for ensuring that electronic components operate safely within their limits, avoiding damage and ensuring effective function.