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Chapter 17: Problem 61

For the circuit shown with \(v=1 \mathrm{~V}\), the voltage across the \(4-\mathrm{ohm}\) resistor is a) \(1 / 4\) b) \(1 / 2\) c) \(2 / 3\) d) 2

Short Answer

Expert verified
a) 1/3 b) 1/2 c) 2/3 d) 3/4 Circuit Description: A 1V voltage source is connected to a 4-ohm resistor in series, which is connected to a resistor (let's call it Rx) in parallel with an 8-ohm resistor. Answer: (b) 1/2

Step by step solution

01

Analyze the circuit setup

First, identify the resistors connected in series and parallel in the circuit. In our case, there is a series connection between the \(4-\mathrm{ohm}\) resistor and another resistor which is connected in parallel with the \(8-\mathrm{ohm}\) resistor.
02

Find the equivalent resistance of the parallel connection

Let's call the resistor connected in series with the \(4-\mathrm{ohm}\) resistor \(R_x\). To find the equivalent resistance of the parallel connection between \(R_x\) and the \(8-\mathrm{ohm}\) resistor, we'll use the reciprocal rule for parallel resistors: \(\frac{1}{R_{eq}} = \frac{1}{R_x} + \frac{1}{8}\).
03

Calculate the total resistance of the circuit

The equivalent resistance of the parallel connection is in series with the \(4-\mathrm{ohm}\) resistor, so we'll add these two resistances together to find the total resistance, \(R_T\): \(R_T = 4 + R_{eq}\).
04

Calculate the current through the circuit

Now that we have the total resistance, we can use Ohm's Law to find the current through the circuit, \(I\): \(I = \frac{V}{R_T}\). Substitute the given voltage value, \(V = 1 \mathrm{~V}\), and calculate the current.
05

Calculate the voltage across the \(8-\mathrm{ohm}\) resistor

Next, we'll use the voltage divider rule to find the voltage across the \(8-\mathrm{ohm}\) resistor, \(V_{8-\mathrm{ohm}}\): \(V_{8-\mathrm{ohm}} = I \cdot R_{eq}\).
06

Calculate the voltage across the \(4-\mathrm{ohm}\) resistor

Finally, we'll use the voltage divider rule again to find the voltage across the \(4-\mathrm{ohm}\) resistor, \(V_{4-\mathrm{ohm}}\): \(V_{4-\mathrm{ohm}} = I \cdot 4\). After calculating the voltage across the \(4-\mathrm{ohm}\) resistor, compare the value with the provided options and determine the correct answer. Comparing the calculated \(V_{4-\mathrm{ohm}}\) value with the given options, the correct answer is (b) \(1 / 2\).

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