Problem 97 - Algebra and Trigonometry

Question

Electrical Resistance If two electrical resistors with resistances \(R_{1}\) and \(R_{2}\) are connected in parallel (see the figure), then the total resistance \(R\) is given by $$ R=\frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}} $$ (a) Simplify the expression for \(R\) (b) If \(R_{1}=10\) ohms and \(R_{2}=20\) ohms, what is the total resistance \(R ?\)

Step-by-Step Solution

Answer

(a) \(R = \frac{R_1 R_2}{R_1 + R_2}\); (b) \(R = 6.67\) ohms.

Step 1: Understanding the Formula for Parallel Resistance

The given formula for the total resistance \( R \) when two resistors \( R_1 \) and \( R_2 \) are connected in parallel is \[R = \frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}.\] Our goal in part (a) is to simplify this expression.

Step 2: Simplifying the Expression for R

Combine the fractions in the denominator: \[\frac{1}{R_1} + \frac{1}{R_2} = \frac{R_2 + R_1}{R_1 R_2}.\] Substitute back into the expression for \( R \): \[R = \frac{1}{\frac{R_2 + R_1}{R_1 R_2}} = \frac{R_1 R_2}{R_1 + R_2}.\] This simplified expression gives the total resistance \( R \) for resistors in parallel.

Step 3: Substitute Given Values for R1 and R2

Now given \( R_1 = 10 \) ohms and \( R_2 = 20 \) ohms. Substitute these values into the simplified formula: \[R = \frac{10 \times 20}{10 + 20}.\]

Step 4: Calculate the Total Resistance

Calculate the numerator: \(10 \times 20 = 200\). Calculate the denominator: \(10 + 20 = 30\). Now substitute back to find \( R \): \[ R = \frac{200}{30}.\] Simplify this to get \( R \approx 6.67 \) ohms.

Key Concepts

Parallel Circuits

Parallel circuits are a fundamental concept in electrical engineering and electronics. Let's break it down so it's easy to understand. In a parallel circuit, two or more components, such as resistors, are connected across the same two points. This means each component has the same voltage applied across it, but the total current in the circuit is the sum of the currents through each component.

Here's why parallel circuits are interesting:

• If any single path gets disconnected, the current will continue to flow through other paths. This is why parallel circuits are used in most household electrical wiring.
• The total resistance in a parallel circuit is always less than the smallest resistor in the circuit. This feature allows for efficient current distribution.

Understanding how components like resistors interact in parallel circuits is crucial for both designing and troubleshooting electrical systems. This concept helps ensure that circuits function safely and effectively in a variety of applications.

Ohm's Law

Ohm's Law is a key principle in the realm of electricity, which relates the voltage ( V ), current ( I ), and resistance ( R ) in an electrical circuit. It's expressed with the formula V = IR . This equation shows us how these three main electrical quantities are interconnected.

Let's break down what each variable represents:

• Voltage ( V ) is the potential difference in charge between two points in the circuit.
• Current ( I ) represents the flow of electric charge.
• Resistance ( R ) is a measure of how much a component resists the flow of charge.

Applying Ohm’s Law in the context of parallel circuits, even though each parallel branch may have a different current flowing through it, each component still experiences the same voltage. Thus, an understanding of Ohm's Law allows us to predict changes in the circuit and calculate unknown values effectively.

Whether designing circuits or analyzing faults, Ohm's Law helps us understand the behavior of electric currents and voltage in practical scenarios.

Resistance Calculation

Calculating resistance can seem tricky, but with the right approach, we can simplify things a lot. When resistors are in parallel, we use a specific formula to find the total resistance:\[ R = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} \]
To simplify this, we can rewrite it as:\[ R = \frac{R_1 R_2}{R_1 + R_2} \]
Using this equation, you can easily find the total resistance of two resistors in parallel without directly dealing with fractions. Let's go through an example to make this clear.

Suppose you have two resistors, one with 10 ohms and another with 20 ohms, connected in parallel. Plug these values into our simplified formula:

• Find the numerator: \( 10 \times 20 = 200 \)
• Find the denominator: \( 10 + 20 = 30 \)
• Calculate the resistance: \( R = \frac{200}{30} \approx 6.67 \) ohms

Notice how the total resistance, \( R \), is less than either individual resistor due to them being in parallel. This approach helps you quickly determine the total resistance in a circuit with components in parallel, which is a vital skill in electronics.