Problem 83 - College Physics

Question

Energy use of home appliances. An 1800 \(\mathrm{W}\) toaster, a 1400 \(\mathrm{W}\) electric frying pan, and a 75 \(\mathrm{W}\) lamp are plugged into the same electrical outlet in a \(20 \mathrm{A}, 120 \mathrm{V}\) circuit. (Note: When plugged into the same outlet, the three devices are in parallel with each other across the 120 \(\mathrm{V}\) outlet.) (a) What current is drawn by each device? (b) Will this combination blow the circuit breaker?

Step-by-Step Solution

Answer

The total current is 27.295 A, which will blow the 20 A circuit breaker.

Step 1: Determine the Current for Each Device

To find the current drawn by each device, use the formula \( I = \frac{P}{V} \), where \( I \) is the current, \( P \) is the power, and \( V \) is the voltage. - For the toaster: \[ I_{toaster} = \frac{1800 \text{ W}}{120 \text{ V}} = 15 \text{ A} \]- For the frying pan: \[ I_{frying pan} = \frac{1400 \text{ W}}{120 \text{ V}} = 11.67 \text{ A} \]- For the lamp: \[ I_{lamp} = \frac{75 \text{ W}}{120 \text{ V}} = 0.625 \text{ A} \]

Step 2: Calculate Total Current Drawn

Since the appliances are in parallel, the total current drawn from the outlet is the sum of the individual currents calculated:\[ I_{total} = I_{toaster} + I_{frying pan} + I_{lamp} = 15 \text{ A} + 11.67 \text{ A} + 0.625 \text{ A} = 27.295 \text{ A} \]

Step 3: Determine If the Circuit Breaker Will Blow

The circuit breaker is rated for 20 A. Since the total current drawn (27.295 A) exceeds 20 A, the breaker would trip to prevent overheating or damage.

Key Concepts

Circuit Analysis

Circuit analysis involves examining an electrical circuit to understand its behavior and to ensure it operates safely and effectively. In this exercise, we analyze a simple home circuit where multiple appliances are connected.
An important part of circuit analysis is understanding how components interact, especially when they share the same power source. In our example, the circuit consists of a toaster, an electric frying pan, and a lamp connected to a 120 V outlet.
By analyzing such circuits, you identify potential issues, like overloading, and ensure the components work within their defined limits. This process helps prevent equipment failure and ensures electrical safety in everyday use.

Parallel Circuits

Parallel circuits have multiple branches, each containing a different electrical component. Each branch connects directly to the voltage source. Here, the toaster, frying pan, and lamp are in parallel, which means each receives the full voltage of 120 V.
In parallel circuits, each device operates independently. If one device is turned off or stops working, the others continue to function. This arrangement is often used in household settings for convenience.
An advantage of parallel circuits is that they maintain a consistent voltage across all devices, ensuring they perform efficiently. However, the total current drawn from the power source can be high, as seen in our exercise where the combined current can potentially overload the circuit.

Current Calculation

To calculate the current for each appliance, we use the formula derived from Ohm's Law: \( I = \frac{P}{V} \), where:

• \( I \) is the current in amperes (A),
• \( P \) is the power in watts (W),
• \( V \) is the voltage in volts (V).

For instance, to find the current for a toaster that uses 1800 W:\( I_{toaster} = \frac{1800 \text{ W}}{120 \text{ V}} = 15 \text{ A} \)
This formula helps us determine how much current each appliance draws, making it easier to ensure circuits are not overloaded. Correct current calculation is critical for preventing potential failures and maintaining safety.

Circuit Breaker

A circuit breaker is a safety device designed to protect an electrical circuit from damage caused by excess current from an overload or short circuit. It interrupts current flow after a fault is detected.
In our exercise, the circuit breaker has a 20 A rating. This value indicates the maximum current it can safely allow before tripping. If the total current exceeds this, the breaker "trips" or shuts off to prevent overheating and reduce fire risk.
Our computed total current of 27.295 A exceeds the 20 A rating, triggering the breaker. This action safeguards the wiring and connected devices by ensuring the circuit does not operate beyond safe limits.

Ohm's Law

Ohm's Law is fundamental in electrical engineering. It describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit: \( V = I \times R \)
While this exercise primarily uses the power formula, Ohm's Law underpins our understanding of these calculations. With the given power and voltage, we deduced the current using a form of this law: \( I = \frac{P}{V} \).
This relationship helps engineers and electricians analyze circuits and design them to operate within safe parameters. Understanding Ohm’s Law allows one to efficiently calculate required values and adapt circuits according to specific needs.