Problem 7 - Applied Physics

Question

What power is used by a heater that has a resistance of \(12.0 \Omega\) and draws a current of \(7.00 \mathrm{~A}\) ?

Step-by-Step Solution

Answer

The power used by the heater is 588.0 W.

Step 1: Understand the Power Formula

Power, represented as \( P \), in an electrical circuit can be calculated using the formula \( P = I^2 R \), where \( I \) is the current in amperes and \( R \) is the resistance in ohms. Our task is to calculate the power consumed by the heater using this formula.

Step 2: Identify the Given Values

From the problem statement, we know the resistance \( R = 12.0 \Omega \) and the current \( I = 7.00 \mathrm{~A} \). These values will be substituted into the power formula.

Step 3: Substitute Values into the Power Formula

Using the formula \( P = I^2 R \), substitute the given values: \( I = 7.00 \mathrm{~A} \) and \( R = 12.0 \Omega \). Thus, \( P = (7.00)^2 \times 12.0 \).

Step 4: Calculate the Power

Perform the calculations: first, calculate \( I^2 = (7.00)^2 = 49.00 \), then multiply by the resistance: \( P = 49.00 \times 12.0 = 588.0 \mathrm{~W} \).

Step 5: Interpret the Result

The power consumed by the heater is \( 588.0 \mathrm{~W} \). This means that the heater uses 588 watts of power when operating with a resistance of 12.0 ohms at a current of 7.00 A.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics and electrical engineering. It describes the relationship between voltage, current, and resistance in an electrical circuit. The formula for Ohm's Law is given by:

• \( V = IR \)
  

where:

• \( V \) is the voltage in volts,
• \( I \) is the current in amperes,
• \( R \) is the resistance in ohms.

This equation tells us that the voltage across a conductor is directly proportional to the current flowing through it, with the resistance being a constant factor. Ohm's Law is crucial because it helps us analyze and design electrical circuits, allowing engineers to predict how a circuit will respond to changes in voltage or resistance.

Power Formula

The Power Formula in electrical engineering is used to calculate the power consumed by an electrical device. Power is represented by \( P \) and is measured in watts (W). One of the ways to express the power equation is:

• \( P = I^2 R \)
  

This formula shows that power is the product of the square of the current (\( I \)) and the resistance (\( R \)). Another common version of the power formula is:

• \( P = VI \)

Here, power is the product of voltage (\( V \)) and current (\( I \)). Utilizing these formulas helps in determining how much electrical energy is being converted into other forms, like heat, light, or motion, in a circuit. In practical applications, knowing the power ensures the safe and efficient operation of electrical devices.

Electrical Resistance

Electrical resistance is a measure of the difficulty faced by current as it flows through a conductor. It is quantified in ohms (\( \Omega \)) and represented by the symbol \( R \). The amount of resistance in an electrical circuit depends on several factors:

• The material of the conductor: Some materials, like copper, have lower resistance, whereas others, like rubber, have higher resistance.
  
• The thickness and length of the conductor: Longer and thinner wires have more resistance compared to shorter and thicker ones.

Resistance plays a crucial role in controlling the flow of current within a circuit. By understanding and managing resistance, engineers can design circuits for specific functions and ensure that they operate safely. Knowing the resistance is also vital for calculating power usage using Ohm's Law and the Power Formula.

Current Measurement

Current measurement is essential for understanding how much charge is flowing in a circuit over a given time. Current, denoted by \( I \) and measured in amperes (A), can be measured using instruments such as ammeters.

• When measuring current, ensure connections are secure, as loose connections can lead to inaccurate readings.
  
• Ammeter devices are essential for diagnostics and circuit troubleshooting because they help identify issues in circuits by showing how current behaves under different conditions.

Accurate current measurement allows for better analysis of electrical systems, ensuring that components are not overloaded and that devices operate within their safe current levels. This understanding is essential for engineers and technicians to maintain efficient and safe electrical systems.