Question

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}.\) a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\) d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\)

Short answer

Answer: Increasing \(R_1\) while keeping \(R_2\) constant will result in a decrease in the effective resistance, \(R\). On the other hand, decreasing \(R_2\) while keeping \(R_1\) constant will result in an increase in the effective resistance, \(R\).

Step-by-step solution

Given the equation \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\), our first task is to solve for \(R\). To do this, solve for the common denominator (\(R_1 R_2\)) and combine the fractions: \[\frac{1}{R} = \frac{R_1 + R_2}{R_1 R_2}\] Now, take the reciprocal of both sides to solve for \(R\): \[R = \frac{R_1 R_2}{R_1 + R_2}\] #Step 2: Differentiate implicitly to find \(\frac{\partial R}{\partial R_1}\) and \(\frac{\partial R}{\partial R_2}\)#

To find the partial derivatives of \(R\) with respect to \(R_1\) and \(R_2\), we use implicit differentiation. First, find \(\frac{\partial R}{\partial R_1}\): \[\frac{\partial R}{\partial R_1} = \frac{(R_2)(R_1 + R_2) - R_1 R_2}{(R_1 + R_2)^2} = -\frac{R_1 R_2}{(R_1 + R_2)^2}\] Next, find \(\frac{\partial R}{\partial R_2}\): \[\frac{\partial R}{\partial R_2} = \frac{(R_1)(R_1 + R_2) - R_1 R_2}{(R_1 + R_2)^2} = \frac{R_1^2}{(R_1 + R_2)^2}\] #Step 3: Describe the effect of increasing \(R_1\) while keeping \(R_2\) constant on \(R\)#

Based on our result from Step 2, we see that the partial derivative \(\frac{\partial R}{\partial R_1}\) is negative. This implies that increasing \(R_1\) while keeping \(R_2\) constant will result in a decrease in the effective resistance, \(R\). In other words, the overall resistance of the circuit will decrease when \(R_1\) is increased. #Step 4: Describe the effect of decreasing \(R_2\) while keeping \(R_1\) constant on \(R\)#

Similarly, we look at the result for the partial derivative \(\frac{\partial R}{\partial R_2}\). We see that it is positive, which means that decreasing \(R_2\) while keeping \(R_1\) constant will result in an increase in the effective resistance, \(R\). In other words, the overall resistance of the circuit will increase when \(R_2\) is decreased.

Key concepts

Implicit Differentiation

Implicit differentiation is a powerful technique used when we have equations where the variables are interdependent and not easily separable. In this context, instead of solving directly for one variable, we differentiate both sides of the equation with respect to a given variable. This allows us to find the derivative of one variable in terms of others.

In the problem presented, we start with the equation for resistors in parallel: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \] Using implicit differentiation helps us find the rate of change of the effective resistance \(R\) with respect to each individual resistor, \(R_1\) and \(R_2\).

By differentiating implicitly, you can differentiate both sides of the equation concerning \(R_1\) and \(R_2\) separately. This involves using the chain rule and recognizing that \(R\) is a function of \(R_1\) and \(R_2\).

For instance, while finding \(\frac{\partial R}{\partial R_1}\), we treat \(R_2\) as a constant and apply the derivative rules accordingly. The process reveals how changes in the values of \(R_1\) or \(R_2\) influence the total resistance \(R\). This approach is especially useful for understanding complex relationships in mathematical models, like electric circuits, where multiple variables interact.

Resistors in Parallel

Resistors in parallel provide a fascinating twist to the way resistance works in circuits. Unlike series circuits, where resistances add up, parallel configurations reduce the total or effective resistance.

When resistors are connected in parallel, each resistor provides an alternative pathway for the current. The effective resistance in parallel (\(R\)) is calculated using the formula: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \] Here, \(R_1\) and \(R_2\) are the resistances of the individual resistors.

The total resistance is always less than the smallest individual resistor in the parallel network. This is because adding more pathways generally allows more current to flow, thereby decreasing the resistance. The effective resistance formula can be rearranged to solve for \(R\) as: \[ R = \frac{R_1R_2}{R_1 + R_2} \] This equation shows how multiple resistors interact to affect the total resistance in a circuit.

This principle is widely used in practical electrical applications where it's desirable to decrease resistance, such as reducing the load on electrical components to prevent overheating.

Effective Resistance

Effective resistance refers to the equivalent resistance of a network of resistors that can be substituted for the real resistors without changing the current or the voltage across the network. It's a way to simplify electrical circuits.

For resistors in parallel, the effective resistance formula derived from: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \] lets us replace two resistors with a single resistor that has the same effect on the circuit's overall behavior.

Understanding changes in effective resistance as one component changes is crucial. For example:

• If \(R_1\) increases while \(R_2\) remains the same, \(R\) decreases. This is shown by the negative partial derivative \(\frac{\partial R}{\partial R_1}\), indicating that the resistance effect reduces overall.
• Conversely, if \(R_2\) decreases while \(R_1\) stays constant, \(R\) increases. The positive partial derivative \(\frac{\partial R}{\partial R_2}\) confirms that reducing this path's resistance increases the system's total resistance.

Effective resistance is a key concept in designing and analyzing electrical circuits. It optimizes how components are arranged to achieve desired electrical properties, allowing efficient energy use across devices.