Question

Resistance \(\quad\) The total resistance \(R\) of two resistors connected in parallel is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) Approximate the change in \(R\) as \(R_{1}\) is increased from 10 ohms to 10.5 ohms and \(R_{2}\) is decreased from 15 ohms to 13 ohms.

Short answer

The short answer will be the numerical value we compute for \(\Delta R\), the approximate change in total resistance \(R\).

Step-by-step solution

Define the Function for Total Resistance

The total resistance \(R\) of two resistors \(R_1\) and \(R_2\) in parallel is given by the formula:\[\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\]So, we have that \(R = \frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}\).

Differentiate Function

Differentiate the equation \(R\) with respect to \(R_1\) and \(R_2\). The partial derivatives are:\[\frac{\partial R}{\partial R_{1}} = \frac{1}{(\frac{1}{R_{1}}+\frac{1}{R_{2}})^{2} \cdot \frac{1}{R_{1}^{2}}}\]and \[\frac{\partial R}{\partial R_{2}} = \frac{1}{(\frac{1}{R_{1}}+\frac{1}{R_{2}})^{2} \cdot \frac{1}{R_{2}^{2}}}\]

Substitute the Initial Values and Compute Derivatives

Substitute \(R_{1}=10 \quad \Omega\) and \(R_{2}=15 \quad \Omega\) into the equations and compute the differentials:\[\frac{\partial R}{\partial R_{1}} = \frac{1} {(\frac{1}{10}+\frac{1}{15})^{2}\cdot \frac{1}{10^{2}}}\]and \[\frac{\partial R}{\partial R_{2}} = \frac{1}{(\frac{1}{10}+\frac{1}{15})^{2}\cdot \frac{1}{15^{2}}}\]

Compute the Approximate Change

The approximate change \(\Delta R\) in \(R\) can be defined as:\[\Delta R \approx (\frac{\partial R}{\partial R_{1}} \cdot \Delta R_{1})+ (\frac{\partial R}{\partial R_{2}} \cdot \Delta R_{2})\]Substitute the calculated differentials and given \(\Delta R_{1}=0.5\) and \(\Delta R_{2}=-2\), and compute \(\Delta R\).

Key concepts

Electrical Resistance

Electrical resistance is a fundamental concept in the study of circuits and electronics. It measures how strongly a material opposes the flow of electric current. The unit of resistance is the ohm (, Ω). The higher the resistance, the more difficult it is for current to flow through a material.

When we talk about resistors in a circuit, we're referring to components specifically designed to provide a certain amount of resistance. This is utilized to control the flow of electric current in various parts of the circuit. In practical applications, it's commonly necessary to calculate the total resistance when multiple resistors are combined in different ways. A key concept here is the difference between resistors in series and parallel: resistors in series add their resistance values directly, but resistors in parallel follow a different rule, which is crucial for calculating their total effective resistance.

Partial Derivatives

Partial derivatives come into play when you're dealing with functions that have more than one variable. In the context of resistance calculation, we have a function that depends on the resistances of two separate resistors. The partial derivative is a tool from calculus that allows you to understand how a function changes as you change just one of its variables, while keeping all other variables constant.

It's like asking the question: If I adjust the resistance of one resistor in the circuit, how much does the overall resistance of the circuit change, without considering changes in the other resistors? Calculating partial derivatives gives us the rates of change for the total resistance with respect to each resistor in isolation. Knowing these rates, we are better equipped to approximate how small changes in individual resistances will affect the total resistance of the parallel circuit.

Resistors in Parallel

The concept of resistors wired in parallel is a fundamental building block of circuit design. Unlike in series circuits, where resistances add up, the total resistance for parallel resistors decreases when more pathways for the current are provided; it's the inversely proportional relationship that's key here. The formula for the total resistance of two resistors in parallel, \(\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}\), reflects how these parallel pathways interact to provide less overall resistance.

Understanding the calculation for resistors in parallel is essential because it allows for precise control over the distribution of current and voltage across various portions of the circuit. This principle is not just theoretical but is applied in everyday objects, from electronic devices to complex electrical grids, underscoring the importance of mastering this calculation for students in electrical engineering and related fields.