Question

An electrical circuit contains three resistors connected in parallel. If these three resistors provide resistance of \(R_{1}, R_{2},\) and \(R_{3}\) ohms, respectively, their combined resistance \(R\) is given by the formula $$ \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} $$ Express \(R\) as a rational expression. Evaluate \(R\) for \(R_{1}=5\) ohms, \(R_{2}=4\) ohms, and \(R_{3}=10\) ohms.

Short answer

R = \frac{20}{11} ohms.

Step-by-step solution

- Understand the Formula

The formula for the combined resistance (R) in a parallel circuit with three resistors (R_1, R_2, and R_3) is given by: \[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\].

- Substitute Variables

Substitute the given values (R_1 = 5, R_2 = 4, R_3 = 10) into the formula: \[\frac{1}{R} = \frac{1}{5} + \frac{1}{4} + \frac{1}{10}\].

- Find Common Denominator

To combine the fractions, find a common denominator. The least common multiple (LCM) of 5, 4, and 10 is 20. Rewrite each fraction with a denominator of 20:\[\frac{1}{5} = \frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20}, \quad \frac{1}{10} = \frac{2}{20}\].

- Add the Fractions

Add the fractions together:\[\frac{1}{R} = \frac{4}{20} + \frac{5}{20} + \frac{2}{20} = \frac{11}{20}\].

- Invert the Fraction

To find R, take the reciprocal of \frac{11}{20}:\[R = \frac{20}{11}\] ohms.

Key concepts

Rational Expression

In mathematics, a rational expression is a fraction where both the numerator and the denominator are polynomials. When dealing with electrical circuits, particularly with parallel resistors, it's crucial to express combined resistance as a rational expression.

This simplifies calculations and helps in analyzing circuit behaviors. In our example, we start with the formula for combined resistance in parallel: \(\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\).
We must rewrite this equation so that \(R\) is on one side by itself. Through mathematical manipulation (finding the least common denominator and adding fractions), we can convert \(\frac{1}{R} = \frac{11}{20}\). Finally, taking the reciprocal, we see that \(R=\frac{20}{11}\). Therefore, \(R\) is expressed as a rational expression \(\frac{20}{11}\).

Common Denominator

In order to add fractions, we need a common denominator. The common denominator helps combine fractions more easily.
Let's use the resistances given: \(R_1=5\), \(R_2=4\), and \(R_3=10\). Calculating parallel resistance involves sums of fractions: \(\frac{1}{5}+\frac{1}{4}+\frac{1}{10}\).
To add these fractions, find their least common multiple (LCM). The LCM of 5, 4, and 10 is 20.
This allows us to rewrite each fraction with a denominator of 20:

• \(\frac{1}{5} = \frac{4}{20}\)
• \(\frac{1}{4} = \frac{5}{20}\)
• \(\frac{1}{10} = \frac{2}{20}\)

Adding these together gives us: \(\frac{4}{20} + \frac{5}{20} + \frac{2}{20} = \frac{11}{20}\). This simplifies the calculation process considerably.

Electrical Circuit Calculations

Understanding electrical circuit calculations is fundamental in physics and engineering. For parallel resistors, we use the formula: \(\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\).

This formula reflects how the overall resistance decreases when more resistors are added in parallel.
In our case, substituting \(R_1=5\), \(R_2=4\), and \(R_3=10\) into the formula gives: \(\frac{1}{R}=\frac{1}{5}+\frac{1}{4}+\frac{1}{10}\).
After finding the common denominator and combining fractions, we get \(\frac{1}{R}=\frac{11}{20}\).
Taking the reciprocal of this fraction, we find the overall resistance \(R=\frac{20}{11}\) ohms.

Understanding these steps and practicing the calculations ensures accuracy in more complex circuit designs and analyses.