Problem 109 - Algebra and Trigonometry

Question

Electrical Resistance If two electrical resistors with resistances R1 and R2 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

R = \( \frac{R_1 R_2}{R_1 + R_2} \); for \( R_1 = 10 \) and \( R_2 = 20 \), \( R \approx 6.67 \) ohms.

Step 1: Write the Formula

We are given the formula for total resistance when two resistors are connected in parallel: \[ R = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} \] Our task is to simplify this expression.

Step 2: Find a Common Denominator

The expression \( \frac{1}{R_1} + \frac{1}{R_2} \) can be combined by finding a common denominator. The common denominator for \( R_1 \) and \( R_2 \) is \( R_1 \times R_2 \). Rewrite the expression as: \[ \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_2}{R_1 R_2} + \frac{R_1}{R_1 R_2} \]

Step 3: Combine the Fractions

Now, combine the fractions: \[ \frac{R_2}{R_1 R_2} + \frac{R_1}{R_1 R_2} = \frac{R_2 + R_1}{R_1 R_2} \]

Step 4: Simplify the Resistance Formula

Substitute the combined fraction into the original resistance formula: \[ R = \frac{1}{\frac{R_1 + R_2}{R_1 R_2}} \] Simplify by multiplying: \[ R = \frac{R_1 R_2}{R_1 + R_2} \] This is the simplified expression for the total resistance \( R \).

Step 5: Insert Values for R1 and R2

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 6: Calculate the Total Resistance

Calculate the expression: \[ R = \frac{200}{30} = \frac{20}{3} \] The total resistance is approximately \( 6.67 \) ohms.

Key Concepts

Parallel Resistors

When we have two or more resistors in an electrical circuit, they can be connected in different ways: series or parallel. Today, we're focusing on parallel connections. In a parallel connection, each resistor is connected across the same two points. This means the voltage across each resistor is the same.
Parallel resistors have an interesting property: their overall resistance is lower than the smallest resistance in the group. This happens because the current has multiple paths to take, reducing the total resistance that the current experiences.
In essence, connecting resistors in parallel helps spread out the electrical flow, making it easier for electricity to travel through the circuit. The formula to calculate total resistance when resistors are placed in parallel is essential to ensure you understand how the total resistance changes with parallel configurations.

Resistor Formula

The formula for calculating the total resistance of resistors in parallel is:

• \[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \]

This formula allows you to calculate the reciprocal of the total resistance by adding up the reciprocals of each individual resistor's resistance.
The formula we used in our exercise simplifies this basic principle for two resistors:

• \[R = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}\]

The beauty of this formula is its capacity to easily extend to more than two resistors, following the same principles of adding their respective reciprocals.

Resistance Calculation

Calculating resistance, especially in parallel circuits, requires attention to detail. The first step involves using the formula above to combine the resistances. Our task was to simplify this for two resistors, resulting in:

• \[ R = \frac{R_1 R_2}{R_1 + R_2} \]

This formula is achieved by combining the fractions and simplifying the equation.
For the example in the problem, we calculated the total resistance for \( R_1 = 10 \) ohms and \( R_2 = 20 \) ohms. Plugging these numbers into our simplified expression resulted in a total resistance of approximately 6.67 ohms.
Understanding how to perform these calculations accurately is key to mastering circuitry, especially in designing circuits where precise resistance levels are crucial.

Ohm's Law

Ohm's Law is a fundamental principle in the study of circuits and electronics. It states that the current \( I \) flowing through a conductor between two points is directly proportional to the voltage \( V \) across the two points and inversely proportional to the resistance \( R \) within the circuit.

• \( V = I \times R \)

This important law helps us understand how voltage, current, and resistance interact. For circuits involving resistors in parallel, knowing how to calculate the overall resistance using our previous formulas allows us to further apply Ohm's Law effectively.
By determining the total resistance in a parallel circuit, we can then use Ohm's Law to find the total current or total voltage, ensuring that we can tackle more complex circuit problems with confidence.