Problem 55 - Physics

Question

The total current delivered to a number of devices connected in parallel is the sum of the individual currents in each device. Circuit breakers are resettable automatic switches that protect against a dangerously large total current by "opening" to stop the current at a specified safe value. A 1650 -W toaster, a 1090 -W iron, and a 1250 -W microwave oven are turned on in a kitchen. As the drawing shows, they are all connected through a 20 -A circuit breaker to an ac voltage of \(120 \mathrm{~V}\). (a) Find the equivalent resistance of the three devices. (b) Obtain the total current delivered by the source and determine whether the breaker will "open" to prevent an accident.

Step-by-Step Solution

Answer

Total current is 33.25 A, breaker will open.

Step 1: Calculate Individual Currents

We will first calculate the individual currents using the formula: \( I = \frac{P}{V} \), where \( P \) is the power and \( V \) is the voltage (120 V in this case).For the toaster:\[ I_{toaster} = \frac{1650}{120} = 13.75 \text{ A} \]For the iron:\[ I_{iron} = \frac{1090}{120} \approx 9.08 \text{ A} \]For the microwave oven:\[ I_{microwave} = \frac{1250}{120} \approx 10.42 \text{ A} \]

Step 2: Calculate Total Current

Add the individual currents to find the total current delivered by the source:\[ I_{total} = I_{toaster} + I_{iron} + I_{microwave} = 13.75 + 9.08 + 10.42 \approx 33.25 \text{ A} \]

Step 3: Determine Breaker Status

Compare the total current to the breaker's safe current limit of 20 A.Since \( I_{total} = 33.25 \text{ A} \) is greater than 20 A, the breaker will "open" to stop the current.

Step 4: Calculate Equivalent Resistance

Using the total current, we will calculate the equivalent resistance \( R_{eq} \) of the three devices, knowing \( V = 120 \text{ V} \) and \( I_{total} = 33.25 \text{ A} \):\[ R_{eq} = \frac{V}{I_{total}} = \frac{120}{33.25} \approx 3.61 \text{ ohms} \]

Key Concepts

Understanding Parallel Circuits

In a parallel circuit, all components are connected alongside each other, forming branches that allow the current to flow through multiple paths. This distinct setup contrasts with series circuits where components are arranged one after the other. One of the standout features of parallel circuits is that the voltage across each component is the same.
Each device in a parallel circuit gets the full voltage from the power source, ensuring uniform operation. This setup is incredibly beneficial in home wiring systems, where numerous devices need to operate independently and efficiently.
One noteworthy characteristic of parallel circuits is that if one pathway is interrupted, the current can continue to flow through the remaining pathways. This characteristic makes them highly advantageous for use in applications where reliability is crucial. Each individual device functions on its own without affecting the others, allowing for more flexible and safer operation.

Steps for Current Calculation in Parallel Circuits

Calculating current in a parallel circuit involves understanding the relationship between power, voltage, and current. The formula \( I = \frac{P}{V} \) helps us determine the current for each device, where \( I \) is the current, \( P \) is the power, and \( V \) is the voltage.
In the context of the kitchen scenario:

• The toaster drawing 1650 W power would have a current \( I_{toaster} \approx 13.75 ext{ A} \).
• Similarly, the iron at 1090 W would draw \( I_{iron} \approx 9.08 ext{ A} \).
• The microwave consuming 1250 W uses \( I_{microwave} \approx 10.42 ext{ A} \).

Subsequently, to find the total current in the circuit, you would sum the currents of all components:
\( I_{total} = I_{toaster} + I_{iron} + I_{microwave} = 33.25 ext{ A} \).
It is crucial to remember that the breaker should handle this total current without surpassing its safety limit. In this instance, the 20-A breaker will trip due to the 33.25 A total current, emphasizing the need for devices to not exceed the circuit breaker's capacity.

Resistance Calculation Simplified

Calculating equivalent resistance in parallel circuits can initially seem daunting, but it's simply about understanding how resistances add up differently than in series circuits. The equivalent resistance \( R_{eq} \) is less than the smallest resistance in the circuit. To find it mathematically when total current and voltage are known, use the formula: \( R_{eq} = \frac{V}{I_{total}} \).
From the earlier problem:
Knowing \( V = 120 ext{ V} \) and \( I_{total} = 33.25 ext{ A} \), applying the formula gives us: \( R_{eq} = \frac{120}{33.25} \approx 3.61 ext{ ohms} \).
This figure confirms that lower equivalent resistance ensures higher efficiency in power delivery typical of parallel circuits. Understanding this principle is key to designing effective electrical systems, ensuring devices operate with maximum stability and distribution of power.