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Chapter 20: Problem 41

Three resistors, \(25,45,\) and \(75 \Omega,\) are connected in series, and a 0.51-A current passes through them. What are (a) the equivalent resistance and (b) the potential difference across the three resistors?

Short Answer

Expert verified
(a) Equivalent resistance is 145 惟. (b) Potential difference is 73.95 V.

Step by step solution

01

Calculate Equivalent Resistance

When resistors are connected in series, the equivalent resistance can be found by adding up the individual resistances. For the resistors given, the equivalent resistance \(R_{eq}\) is: \(R_{eq} = 25 \, \Omega + 45 \, \Omega + 75 \, \Omega\).
02

Add the Resistances

Perform the addition: \(R_{eq} = 25 + 45 + 75 = 145 \, \Omega\). Thus, the equivalent resistance of the series combination is \(145 \, \Omega\).
03

Identify Total Current in Series

In a series circuit, the same current flows through each resistor. Given that the current is \(0.51 \, \text{A}\), we can use this information to find the potential difference across the series connected resistors.
04

Calculate Potential Difference across Resistors

Use Ohm's Law, \(V = IR\), to find the potential difference across the equivalent resistance. Substitute \(I = 0.51 \, \text{A}\) and \(R_{eq} = 145 \, \Omega\) into the equation: \(V = 0.51 \, \text{A} \times 145 \, \Omega\).
05

Compute the Result

Calculate the value:\(V = 0.51 \times 145 = 73.95 \, \text{V}\). Therefore, the potential difference across the three resistors is \(73.95 \, \text{V}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equivalent Resistance
In a series circuit, determining the equivalent resistance is a straightforward process. You simply add up the values of all the resistors in the circuit. This is because in a series connection, the current must pass through each resistor one after the other. Each resistor adds to the total opposition to the flow of current.
For example, if you have resistors of 25 惟, 45 惟, and 75 惟, the equivalent resistance can be calculated as follows:
  • Sum the resistances: \(R_{eq} = 25 \, \Omega + 45 \, \Omega + 75 \, \Omega\)
  • This results in an equivalent resistance of \(145 \, \Omega\)
Thus, the entire series circuit behaves as if it has a single resistor of \(145 \, \Omega\). This simplification can make analyzing the circuit much easier.
Ohm's Law
Ohm's Law is an essential principle in electronics. It defines the relationship between current, voltage, and resistance. The formula is \(V = IR\), where \(V\) is the voltage (potential difference), \(I\) is the current, and \(R\) is the resistance.
This law tells us that the voltage across a circuit is directly proportional to the current flowing through it and the resistance it encounters. In a practical situation, it allows you to calculate one variable if you have the other two.
  • For instance, with a current \(I = 0.51 \, \text{A}\) and equivalent resistance \(R_{eq} = 145 \, \Omega\), you can calculate the voltage:
  • Substitute into the formula: \(V = 0.51 \, \text{A} \times 145 \, \Omega\)
  • This results in \(V = 73.95 \, \text{V}\)
Ohm鈥檚 Law is crucial because it helps you understand how changes in resistance or current affect the overall behavior of the circuit.
Potential Difference
The potential difference, often referred to as voltage, is a measure of the electrical energy difference between two points in a circuit. It drives the current from one point to another.
In a series circuit, finding the total potential difference involves understanding that this voltage is evenly spread across each resistor in proportion to their resistance. Using Ohm鈥檚 Law, you can find the potential difference across the entire series of resistors or each resistor individually.
  • With our example, for a current of 0.51 A flowing through and a total resistance of 145 惟, the total potential difference is \(73.95 \, \text{V}\).
  • This voltage ensures that the current overcomes the total resistance offered by all resistors in series.
The concept of potential difference is vital as it helps in determining how effective a circuit can be in doing work, like lighting a bulb or powering a device.

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Most popular questions from this chapter

A wire has a resistance of \(21.0 \Omega .\) It is melted down, and from the same volume of metal a new wire is made that is three times longer than the original wire. What is the resistance of the new wire?

The resistance of a bagel toaster is \(14 \Omega .\) To prepare a bagel, the toaster is operated for one minute from a \(120-\mathrm{V}\) outlet. How much energy is delivered to the toaster?

The drawing shows two capacitors that are fully charged \(\left(C_{1}=2.00 \mu \mathrm{F}, q_{1}=6.00 \mu \mathrm{C} ; C_{2}=\right.\) \(\left.8.00 \mu F, q_{2}=12.0 \mu C\right) .\) The switch is closed, and charge flows until equilibrium is reestablished (i.e., until both capacitors have the same voltage across their plates). Find the resulting voltage across either capacitor.

An electric blanket is connected to a \(120-\mathrm{V}\) outlet and consumes \(140 \mathrm{W}\) of power. What is the resistance of the heater wire in the blanket?

A coffee cup heater and a lamp are connected in parallel to the same \(120-\mathrm{V}\) outlet. Together, they use a total of \(111 \mathrm{W}\) of power. The resistance of the heater is \(4.0 \times 10^{2} \Omega .\) Find the resistance of the lamp.
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