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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

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?

Show that the power supplied to the circuit in the figure by the battery with internal resistance is maximum when the resistance of the resistor in the circuit, \(R\), is equal to \(R_{i}\). Determine the power supplied to \(R\). For practice, calculate the power dissipated by a \(12.0-\mathrm{V}\) battery with an internal resistance of \(2.00 \Omega\) when \(R=1.00 \Omega, R=2.00 \Omega,\) and \(R=3.00 \Omega\)

The most common material used for sandpaper, silicon carbide, is also widely used in electrical applications. One common device is a tubular resistor made of a special grade of silicon carbide called carborundum. A particular carborundum resistor (see the figure) consists of a thick-walled cylindrical shell (a pipe) of inner radius \(a=\) \(1.50 \mathrm{~cm},\) outer radius \(b=2.50 \mathrm{~cm},\) and length \(L=60.0 \mathrm{~cm} .\) The resistance of this carborundum resistor at \(20 .{ }^{\circ} \mathrm{C}\) is \(1.00 \Omega\). a) Calculate the resistivity of carborundum at room temperature. Compare this to the resistivities of the most commonly used conductors (copper, aluminum, and silver). b) Carborundum has a high temperature coefficient of resistivity: \(\alpha=2.14 \cdot 10^{-3} \mathrm{~K}^{-1} .\) If, in a particular application, the carborundum resistor heats up to \(300 .{ }^{\circ} \mathrm{C},\) what is the percentage change in its resistance between room temperature \(\left(20 .{ }^{\circ} \mathrm{C}\right)\) and this operating temperature?

When a battery is connected to a \(100 .-\Omega\) resistor, the current is \(4.00 \mathrm{~A}\). When the same battery is connected to a \(400 .-\Omega\) resistor, the current is 1.01 A. Find the emf supplied by the battery and the internal resistance of the battery.

The Stanford Linear Accelerator accelerated a beam consisting of \(2.0 \cdot 10^{14}\) electrons per second through a potential difference of \(2.0 \cdot 10^{10} \mathrm{~V}\) a) Calculate the current in the beam. b) Calculate the power of the beam. c) Calculate the effective ohmic resistance of the accelerator.
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