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Chapter 25: Problem 42

A resistor of unknown resistance and a \(35-\Omega\) resistor are connected across a \(120-\mathrm{V}\) emf device in such a way that an 11 -A current flows. What is the value of the unknown resistance?

Short Answer

Expert verified
Answer: The value of the unknown resistor is approximately 16 Ω.

Step by step solution

01

Determine the type of connection

Since the total current flowing through the circuit is given, this implies that the resistors are connected in parallel. In a parallel connection, the total current is the sum of the currents flowing through each branch of the resistors.
02

Apply Ohm's Law to the known resistor

Ohm's Law states that the voltage across a resistor (\(V_R\)) is equal to the product of the current through the resistor (\(I_R\)) and the resistor's resistance (\(R\)). In mathematical form: \(V_R = I_RR\). We apply Ohm's Law to the 35-Ω resistor to find its current. Let's call this current \(I_1\). \(120 \mathrm{V} = I_1 (35 \Omega)\) Solving for \(I_1\), we get: \(I_1 = \dfrac{120 \mathrm{V}}{35 \Omega} = 3.43 \,\text{A}\)
03

Find the current through the unknown resistor

Since the total current in the circuit is 11-A, we can find the current through the unknown resistor by subtracting the current through the 35-Ω resistor from the total current. Let's call the current through the unknown resistor \(I_2\). \(I_2 = 11 \,\text{A} - 3.43 \,\text{A} = 7.57 \,\text{A}\)
04

Apply Ohm's Law again to find the unknown resistance

Now, we apply Ohm's Law to the unknown resistor, using the calculated current \(I_2\) and the given voltage of the emf device. Let's call the unknown resistance \(R_2\). \(120 \mathrm{V} = 7.57 \,\text{A}R_2\) Solving for \(R_2\), we get: \(R_2= \dfrac{120\mathrm{V}}{7.57 \,\text{A}} = 15.85 \,\Omega\)
05

Round off the answer to an appropriate number of significant figures

Given values in this exercise have two significant figures; hence the result should be rounded off to two significant figures. So, the value of the unknown resistance is approximately \(16\,\Omega\).

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Most popular questions from this chapter

You are given two identical batteries and two pieces of wire. The red wire has a higher resistance than the black wire. You place the red wire across the terminals of one battery and the black wire across the terminals of the other battery. Which wire gets hotter?

A copper wire that is \(1 \mathrm{~m}\) long and has a radius of \(0.5 \mathrm{~mm}\) is stretched to a length of \(2 \mathrm{~m}\). What is the fractional change in resistance, \(\Delta R / R,\) as the wire is stretched? What is \(\Delta R / R\) for a wire of the same initial dimensions made out of aluminum?

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A battery has a potential difference of \(14.50 \mathrm{~V}\) when it is not connected in a circuit. When a \(17.91-\Omega\) resistor is connected across the battery, the potential difference of the battery drops to \(12.68 \mathrm{~V}\). What is the internal resistance of the battery?

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