Question

Three resistors are connected to a power supply with \(V=110 . \mathrm{V}\) as shown in the figure a) Find the potential drop across \(R_{3}\) b) Find the current in \(R_{1}\). c) Find the rate at which thermal energy is dissipated from \(R_{2}\).

Short answer

The formula to calculate the equivalent resistance of resistors connected in parallel is: $$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$$

Step-by-step solution

Analyze the circuit

First, let's identify the type of connections for the resistors. \(R_1\) and \(R_3\) are connected in series, and this series connection is in parallel with \(R_2\).

Calculate the equivalent resistance

To find the potential drop across \(R_3\) and the current in \(R_1\), we need to find the equivalent resistance of the parallel connection, which we will call \(R_{eq}\). In order to find \(R_{eq}\), we first need to find the equivalent resistance of the series connection of \(R_1\) and \(R_3\). The equivalent resistance of resistors in series is given by: $$R_{eq1} = R_1 + R_3$$ Now we can find the equivalent resistance of \(R_{eq1}\) and \(R_2\) in parallel using the formula: $$\frac{1}{R_{eq}} = \frac{1}{R_{eq1}} + \frac{1}{R_2}$$

Find the total current coming from the power supply

Now that we have found \(R_{eq}\), we can use Ohm's law to find the total current coming from the power supply. Ohm's law states that: $$I = \frac{V}{R}$$ Here, the total current \(I\) is equal to the voltage \(V\) divided by the equivalent resistance \(R_{eq}\). Substitute the values and calculate the current.

Find the potential drop across \(R_3\)

To find the potential drop across \(R_3\), first, we need to find the current flowing through \(R_1\) and \(R_3\). Since both are in series, the current flowing through them is the same as the total current coming from the power supply. After finding the current flowing through \(R_3\), use the Ohm's law formula to find the potential drop across it: $$V_{R_3} = I_{R_3} \times R_3$$

Find the current flowing through \(R_1\)

As mentioned earlier, the current flowing through \(R_1\) is the same as the total current coming from the power supply. Therefore, the current in \(R_1\) is equal to the current calculated in step 3.

Calculate the rate of thermal energy dissipation

To find the rate at which thermal energy is dissipated from \(R_2\), we first need to find the current flowing through \(R_2\). Since \(R_2\) is in parallel with the series connection of \(R_1\) and \(R_3\), the current flowing through it can be found using the formula: $$I_{R_2} = \frac{V}{R_2}$$ Next, use the formula for power to find the rate of thermal energy dissipation: $$P_{R_2} = I_{R_2}^2 \times R_2$$

Key concepts

Series and Parallel Circuits

In electrical engineering, circuits are often composed of various elements such as resistors, capacitors, and inductors connected in a variety of ways. Two of the most fundamental types of connections are series and parallel circuits.

**Series Circuits:** In a series circuit, components are connected end-to-end, so there is only a single path for the current to follow. The same current flows through each component, but the voltage across each component can be different. The total resistance in a series circuit is simply the sum of the individual resistances. For example, if resistors with resistances of \(R_1\) and \(R_3\) are connected in series, their combined resistance would be \(R_{eq1} = R_1 + R_3\).

**Parallel Circuits:** In parallel circuits, components are connected across the same two points, creating multiple paths for the current. As a result, the voltage across each component is the same, but the current can vary. The equivalent resistance in a parallel circuit is a bit more complex and can be found using the formula \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots\), where \(R_{eq}\) is the total equivalent resistance of the parallel network.

Understanding the differences between series and parallel circuits is crucial for analyzing and solving various electrical circuit problems. It helps in predicting how circuits will behave when voltages and currents are applied.

Equivalent Resistance

Equivalent resistance is a concept used to simplify complex resistor networks into simpler forms, making analysis easier.

When resistors are connected in series, the equivalent resistance is straightforward: it is the sum of all individual resistances. This is because the current has only one path to follow, adding up resistive effects of each element in line.

On the other hand, finding the equivalent resistance in a parallel connection involves more calculations. The formula \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}\) or its general form for more resistors, becomes crucial. In parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. This is due to current having multiple paths, therefore reducing the overall resistance as each additional path adds another route for current flow.

To find the overall equivalent resistance in circuits that combine both series and parallel elements, one must first simplify series components, then tackle parallel sections. This iterative process continues until you're left with a single equivalent resistance.

Grasping the concept of equivalent resistance allows for the application of Ohm's Law to determine currents and voltages at various points within a circuit.

Thermal Energy Dissipation

Components like resistors convert electrical energy into thermal energy, which is dissipated as heat within them. This process, known as thermal energy dissipation, is crucial for understanding both the efficiency and safety of electrical circuits.

**Power Dissipation**: The rate at which a resistor dissipates energy can be found using several formulations of the power equation. One useful form is \(P = I^2 \times R \), where \(I\) is the current flowing through the resistor, and \(R\) is its resistance. This equation highlights that power dissipation depends on the square of the current, indicating that small increases in current can lead to significant increases in thermal energy loss.

**Effects of Power Dissipation**: Excessive thermal dissipation can lead to overheating, which may damage components or create fire hazards. Thus, calculating and managing power dissipation is essential in circuit design and application.

By carefully selecting resistor values and considering the paths of current within a circuit, we can control the rate of energy dissipation to protect the circuit and ensure efficient functioning.

Understanding how thermal energy is dissipated helps in choosing appropriate components and designing circuits that stay within safe operational limits.