Question

A \(2580-\Omega\) resistor \(R_{2}\) is wired in parallel with a resistor \(R_{1} .\) Graph the equivalent resistance \(R\) for values of \(R_{1}\) from 0 to \(5000 \Omega\). Use the equation $$ R=\frac{R_{1} R_{2}}{R_{1}+R_{2}} $$

Short answer

Plot a curve on a graph with the x-axis as \( R_{1} \) ranging from 0 to 5000 \( \Omega \) and the y-axis as the equivalent resistance \( R \). The equivalent resistance \( R \) decreases as \( R_{1} \) increases and approaches \( R_{2} = 2580 \Omega \) as \( R_{1} \) reaches very large values.

Step-by-step solution

Understand the Equation

The equivalent resistance of two resistors in parallel is given by the formula: \( R = \frac{R_{1} R_{2}}{R_{1} + R_{2}} \). The equation is derived from the fact that for parallel resistors, the reciprocal of the total resistance (1/R) is the sum of the reciprocals of the individual resistances (1/R1 + 1/R2).

Set Up the Graph Axes

On graph paper or using a graphing utility, draw a set of Cartesian axes. Label the horizontal axis as \( R_{1} \) with values from 0 to 5000 \( \Omega \). Label the vertical axis as the equivalent resistance \( R \), choosing an appropriate scale to accommodate the expected range of values for \( R \).

Calculate Data Points

Calculate multiple points for the graph by choosing different values for \( R_{1} \) within the given range and computing the corresponding equivalent resistance \( R \) using the formula. It's beneficial to use a range of values including 0, the value of \( R_{2} = 2580 \Omega \), and 5000 \Omega, as well as values in between.

Plot the Points

For each calculated \( R_{1} \) and \( R \), plot the point on the graph. The point represents the equivalent resistance when \( R_{1} \) has a specific value. Connect the plotted points with a smooth curve to illustrate how the equivalent resistance changes with \( R_{1} \).

Analyze the Curve

Once all points are plotted and the curve is drawn, we can observe how the equivalent resistance decreases as \( R_{1} \) increases. It is important to note that as \( R_{1} \) approaches infinity, the equivalent resistance \( R \) approaches \( R_{2} \), and as \( R_{1} \) approaches 0, the equivalent resistance also approaches 0.

Key concepts

Resistors in Parallel

When electrical circuit designers and hobbyists work with resistors, they often combine them in parallel to achieve a desired level of resistance. This configuration is where two or more resistors are connected across the same two points, creating multiple paths for current to flow. Unlike resistors in series where resistances add up, resistors in parallel provide a way to reduce the total or equivalent resistance of a circuit.

The unique aspect of parallel resistors is that the total (equivalent) resistance is always less than the smallest resistor in the network. This is due to the increase in the total current carrying area presented by the resistors combined. If you have two resistors in parallel, current can flow through both simultaneously, effectively doubling the pathway for current when compared with just one resistor and hence halving the resistance.

Equivalent Resistance Calculation

Calculating the equivalent resistance of resistors in parallel involves a specific formula: \( R = \frac{R_{1}R_{2}}{R_{1} + R_{2}} \). This equation might look a bit intimidating at first, but it's based on a simple principle: the inverse of the total resistance for parallel resistors equals the sum of their inverses. Here's a simpler way to visualize it – imagine each resistor as a road. The more roads cars can take at the same time, the less congested each road becomes.

To solve for the equivalent resistance, you take the product of the resistances and divide it by their sum. This calculation becomes increasingly complex with more resistors, but for just two, it is fairly straightforward. A tip for students: always remember to check your units! Inconsistencies here can lead to errors in your calculations.

Graphing Techniques in Electronics

Graphing is a powerful technique in electronics used to visualize the relationship between different variables, such as resistance, voltage, and current. The step-by-step solution of plotting the equivalent resistance against various values of \(R_{1}\) provides an illustrative method to understand and predict how changes in one component affect the overall circuit.

Using graphing, we can easily see trends, such as the curve that shows the decreasing nature of resistance with increasing \(R_{1}\). In electronics, such graphical analysis is crucial not only in theoretical exercises but also in practical applications, like interpreting the behavior of sensors or the output of a circuit under different conditions. In education, graphing can transform abstract equations into concrete visual knowledge, aiding in deeper comprehension of electronic principles.

Ohm's Law Applications

Ohm's Law, which states that \(V = IR\) (voltage equals current times resistance), is a fundamental principle used throughout electronics to analyze circuits. When applying Ohm's Law to parallel resistors, we consider that the voltage across each resistor is the same. This insight can help solve for unknown variables in complex networks, and it is directly related to the concept of equivalent resistance.

For example, if you know the voltage across parallel resistors and the equivalent resistance, you can determine the total current using Ohm's Law. Conversely, if you know the total current and the individual resistances, you can calculate the voltage. Understanding how to apply Ohm's Law in the context of equivalent resistance serves as an essential skill for anyone studying electronics or electrical engineering.