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Chapter 25: Problem 75

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

Expert verified
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

01

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 \(R_2\).
02

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: $$\frac{1}{R_{eq}} = \frac{1}{R_{eq1}} + \frac{1}{R_2}$$
03

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.
04

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: $$V_{R_3} = I_{R_3} \times R_3$$
05

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.
06

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: $$P_{R_2} = I_{R_2}^2 \times R_2$$

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Most popular questions from this chapter

Two conductors of the same length and radius are connected to the same emf device. If the resistance of one is twice that of the other, to which conductor is more power delivered?

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