Question

Two resistors with resistances \(R_{1}\) and \(R_{2}\) are connected in parallel. Demonstrate that, no matter what the actual values of \(R_{1}\) and \(R_{2}\) are, the equivalent resistance is always less than the smaller of the two resistances,

Short answer

Prove your answer. Answer: Yes, the equivalent resistance in a parallel circuit with two resistors is always less than the smaller resistance. This can be proved using the formula for equivalent resistance in a parallel connection, \(R_{eq} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\). By assuming \(R_1 \leq R_2\) and rearranging the inequality, we get \(0 < R_{1}^2\), which always holds true as the square of a resistance (a positive value) is greater than zero.

Step-by-step solution

Formula for equivalent resistance of parallel resistors

In a parallel circuit, the equivalent resistance \(R_{eq}\) can be calculated using the following formula: \[\frac{1}{R_{eq}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}\]

Rewrite the formula and solve for \(R_{eq}\)

We will rewrite the formula to express the equivalent resistance \(R_{eq}\) in terms of the individual resistances \(R_1\) and \(R_2\). To achieve this, let's add the fractions on the right side and solve for \(R_{eq}\): \[\frac{1}{R_{eq}} = \frac{R_{1}+R_{2}}{R_{1}R_{2}}\] Now, to obtain \(R_{eq}\), we will take the reciprocal of both sides of the equation: \[R_{eq} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\]

Prove that \(R_{eq}\) is always less than the smaller resistance

With our present equation, we will show that \(R_{eq}\) is less than the smaller of the two resistances, \(R_1\) and \(R_2\). Without loss of generality, let us assume that \(R_1 \leq R_2\). The inequality we want to prove is: \[R_{eq} = \frac{R_{1}R_{2}}{R_{1}+R_{2}} < R_{1}\] We will take a cross-multiplication approach to simplify and rearrange the inequality: \[R_{1}R_{2} < R_{1}(R_{1} + R_{2})\] Now, let's distribute \(R_{1}\) to the terms in the parentheses on the right side of the inequality: \[R_{1}R_{2} < R_{1}^2 + R_{1}R_{2}\] Subtracting \(R_{1}R_{2}\) from both sides of the inequality gives: \[0 < R_{1}^2\] Since it is always true that the square of a resistance (a positive value) is greater than zero, the inequality holds, and we have demonstrated that, no matter the actual values of \(R_{1}\) and \(R_{2}\), the equivalent resistance in a parallel connection is always less than the smaller of the two resistances.

Key concepts

Resistance in Parallel

When dealing with resistance in parallel circuits, it is crucial to understand how they differ from series circuits. In a parallel configuration, multiple paths for the current are created as each resistor is connected separately to the voltage source. This setup influences the total resistance in a way that might seem counterintuitive at first: the combined resistance drops.

To calculate the equivalent resistance (\(R_{eq}\)) of two such resistors, we use the formula: \[\frac{1}{R_{eq}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}\]. By finding a common denominator and adding these fractions, it becomes apparent why \(R_{eq}\) is always lower than either \(R_{1}\) or \(R_{2}\), especially the smallest of them. The simultaneous paths allow electrons to flow more easily compared to having to pass through either resistance alone, which is similar to opening more lanes on a highway; traffic flows better, resulting in lower overall resistance.

Ohm's Law

Ohm’s law is a foundational principle used in circuit analysis. It states a direct proportionality between the voltage across a conductor and the current through it, provided the temperature remains constant. Mathematically, Ohm's law is expressed as \( V = IR \), where \(V\) is the voltage, \(I\) the current, and \(R\) the resistance.

This principle is instrumental when analyzing parallel circuits, as it helps determine the current distribution across the resistors. When coupled with the formula for equivalent resistance, it becomes a powerful tool to analyze various aspects of a circuit, such as voltage drops and power consumption, ensuring each component's operation is understood and effectively predicted.

Circuit Analysis

Circuit analysis is a systematic approach to understanding the behaviors and relationships within an electrical circuit. By applying various principles, such as Ohm’s law and Kirchhoff’s laws, one can deduce how current, voltage, and resistance interact in complex networks.

When finding equivalent resistance in parallel circuits, circuit analysis involves a set of steps that begins with identifying all parallel relationships and then applying the relevant formulas to simplify the circuit to its basics. The goal of such simplification is to create a manageable number of components that can correctly represent the original circuit's behavior. By doing so, predictions can be made about system responses to changes, and optimal designs for desired electrical behavior can be achieved.

Electrical Resistance

Electrical resistance is a measure of the opposition to the flow of electric current within a conductor. The unit of resistance is the ohm, symbolized by \(\Omega\). Essentially, electrical resistance quantifies how much a material reduces the current flow, acting somewhat like a choke point.

Materials with high resistance, such as rubber and glass, are called insulators because they impede the flow of electrons. Conversely, conductors have low resistance and allow current to flow easily. Integrity of materials, temperature, and physical dimensions all impact resistance. Through careful material selection and circuit design, engineers control the amount of resistance in a circuit to safeguard components, manage energy consumption, and ensure stability and efficiency of the electrical system.