Problem 50 - Introductory Algebra for College Students

Question

If two electrical resistors with resistances \(R_{1}\) and \(R_{2}\) are connected in parallel (see the figure), then the total resistance in the circuit is given by the complex rational expression $$\frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}.$$ Simplify the expression. Then find the total resistance if \(R_{1}=10\) ohms and \(R_{2}=20\) ohms.

Step-by-Step Solution

Answer

The total resistance of the circuit is \(20/3\) ohms.

Step 1: Simplification of the Complex Rational Expression

Start by rationalizing the denominator to remove the complex fraction. The expression turns into: \( (\frac{1}{R_{1}}+\frac{1}{R_{2}})^{-1} \). Then rewrite the expression as the product of reciprocals: \( R_{1}R_{2}/(R_{1}+R_{2}) \).

Step 2: Calculation of the Total Resistance

Now use the formula obtained in step 1 for \(R_{1}=10\) ohms and \(R_{2}=20\) ohms. Substituting these values into the formula, you get: \( 10 * 20 / (10 + 20) \) which simplifies to \( 200/30 = 20/3 \) ohms.

Key Concepts

Electrical Resistors

Electrical resistors are fundamental components in circuits. Their primary role is to limit or regulate the flow of electrical current. In essence, resistors disappoint the passage of electricity, converting it into heat. Each resistor is characterized by its resistance, measured in ohms (Ω), which tells you how much it resists the flow of electricity. In circuits, resistors can be combined in different configurations. The two basic configurations are series and parallel. Understanding these configurations is essential for effectively calculating total resistance in any circuit arrangement. When resistors are connected, the way they impact current flow changes, which can alter the total resistance experienced by the circuit.

Resistance Calculation

Calculating resistance properly involves understanding how resistors affect circuits. In the context of parallel circuit configurations, each resistor provides an additional path for current, changing the total resistance calculation.When resistors - are placed in parallel, - the total resistance decreases.To find the total resistance of two parallel resistors with resistances \( R_1 \) and \( R_2 \), the formula used is:\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]This formula is rooted in the laws of electrical circuits, where the reciprocal of the total resistance is equivalent to the reciprocal sum of individual resistances. By simplifying the complex fraction from the problem, we discover a practical approach to find total resistance:\[ R_{total} = \frac{R_1 \times R_2}{R_1 + R_2} \]This helps us calculate quickly and clearly, without having to solve a complex fraction manually every time.

Parallel Circuits

Parallel circuits are everywhere, from household wiring to complex electronics. Unlike series circuits, where components are arranged in a single path, parallel circuits consist of multiple paths for current. In parallel circuits: - Each component receives the same voltage. - Current may vary along each path. - Adding more resistors reduces overall resistance, as the current has additional paths. The beauty of parallel circuits lies in their reliability. If one path fails, current can continue through other routes. Plus, by adding more resistors, you can manage the current more efficiently despite lowering resistance. This characteristic is particularly beneficial in evenly distributing loads and ensuring consistent voltage across all components. Understanding how resistors work within these configurations is crucial for designing stable and efficient electrical systems.