Question

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?

Short answer

(a) The simplified expression is$R=\frac{R_1{R}_2}{R_2+R_1}$

(b) The total resistance is about 6.67 ohms.

Step-by-step solution

Part a. Step 1. Apply the concept of addition of rational expressions.

To add or subtract rational expressions, we first find a common denominator and then use the following property of fractions:

$\frac{A}{C}+\frac{B}{C}=\frac{A+B}{C}$

The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

Part a. Step 2. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Part a. Step 3. Analyze the expression.

Given

$R=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}$

The LCD is$R_1{R}_2$

Part a. Step 4. Simplify the expression.

Consider$R=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}$

So, the simplified expression is given by:

$\begin{array}{l}R=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\\ \Rightarrow R=\frac{1}{\frac{R_2+R_1}{R_1{R}_2}}\\ \Rightarrow R=\frac{R_1{R}_2}{R_2+R_1}\end{array}$

Part b. Step 1. Apply the concept of addition of rational expressions.

To add or subtract rational expressions, we first find a common denominator and then use the following property of fractions:

$\frac{A}{C}+\frac{B}{C}=\frac{A+B}{C}$

The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

Part b. Step 2. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Part b. Step 3. Use part (a).

From part (a), we have

$R=\frac{R_1{R}_2}{R_2+R_1}$

Part b. Step 4. Total Resistance.

Consider $R=\frac{R_1{R}_2}{R_2+R_1}$

Substitute the values $R_1=10,R_2=20$:

$\begin{array}{l}R=\frac{R_1{R}_2}{R_2+R_1}\\ \Rightarrow R=\frac{\left(10\right)\left(20\right)}{20+10}\\ \Rightarrow R=\frac{200}{30}\\ \Rightarrow R=\frac{20}{3}\approx 6.67\end{array}$

So, the total resistance is about 6.67 ohms.