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Chapter 10: Problem 10

A circuit consists of a \(3.0\)-volt battery and two resistors connected in series with it. The first resistor has a resistance of 10 . ohms. The second has a resistance of 15 . ohms. Find the current in the circuit. (Ans. 0.12 A.)

Short Answer

Expert verified
The current in the circuit is \(0.12 \, \text{A}\).

Step by step solution

01

Calculate the Total Resistance

Adding the resistance of the first resistor (10 ohms) and the resistance of the second resistor (15 ohms), the total resistance \( R \) is given by \( R = 10 \, \text{ohms} + 15 \, \text{ohms} = 25 \, \text{ohms} \)
02

Apply Ohm's Law

Using Ohm's law, which says that the current \( I \) in a circuit is the voltage \( V \) divided by the resistance \( R \), we have \( I = \frac{V}{R} \). Substitute \( V = 3.0 \, \text{V} \) and \( R = 25 \, \text{ohms} \) into the equation.
03

Solve for Current

Upon substituting the values into Ohm's law, you get \( I = \frac{3.0\, \text{V}}{25 \, \text{ohms}} \). Solving the numerator by the denominator, we get 0.12 Amps.

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Key Concepts

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

Circuit Analysis
Circuit analysis is the process of understanding a circuit's behavior by examining its components and how they interact. In this exercise, we have a simple circuit consisting of a battery and resistors connected in series. This means the electric current flows through each resistor one after the other.

To analyze circuits, we need to determine two main things: total resistance and current flow. With these, we can predict how the circuit operates under different conditions. Knowing these factors allows for better circuit design and problem-solving. In this case, analyzing the circuit helps find the current using the known voltage and resistances.
Resistors in Series
Resistors in series are arranged such that the current flows through each resistor sequentially. This arrangement means the total resistance increases with each added resistor. The formula for total resistance \( R_t \) in series is the sum of all individual resistances:

\[ R_t = R_1 + R_2 + \, ... \, \]

In the exercise, two resistors with resistances of \(10\, \Omega\) and \(15\, \Omega\) are in series, making the total resistance \( R_t = 25 \, \Omega\). This addition of resistances helps us understand how easily current can flow through the circuit.
  • Current is the same through all components in series.
  • Increased resistance means less current flow for a given voltage.
Voltage and Current
Voltage and current are key concepts in understanding how circuits work. Voltage (measured in volts) is the electric potential difference between two points, acting like a force pushing electrons through a circuit. Current (measured in amperes) is the flow rate of these electrons.

Ohm's Law, expressed as \( I = \frac{V}{R} \), relates voltage (\( V \)), current (\( I \)), and resistance (\( R \)). Knowing any two of these allows us to solve for the third. In our circuit:
  • Voltage is \( 3.0 \, V \) from the battery.
  • Total resistance is \( 25 \, \Omega \) from the resistors in series.
  • Using Ohm's law, the current is found to be \( 0.12 \, A \).
Understanding these relationships is crucial for predicting how different elements in a circuit will interact.

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Most popular questions from this chapter

An automobile's 12 -volt battery is used to drive a starter motor, which for several seconds draws a power of \(3.0 \mathrm{~kW}\) from the battery. If the motor can be modeled by a single resistor, what is the current in the motor circuit while the motor is operating? (Ans. 250 A.)

An automobile's 12 -volt battery is used to light the automobile's two headlights. Each headlight can be modeled as a \(1.00\)-ohm resistor. If the two headlights are hooked up to the battery in series to form a circuit, what is the power produced in each headlight? Why would you not wire car lights in series?

A circuit consists of a \(12.0\)-volt battery and a resistor connected in series with it. The current is 105 . A. Find the resistance.

An automobile's 12 -volt battery is used to light the automobile's two headlights. Each headlight can be modeled as a single resistor. If the two headlights are hooked up to the battery in parallel to form a circuit and each headlight is to produce a power of 100 . W, what should the resistance of each headlight be? (Ans. 1.4 ohms per bulb.)

Suppose a car has headlights operating in parallel but with different resistances-one of \(2.0 \mathrm{ohms}\) and the other of \(3.0 \mathrm{ohms}\). Suppose the headlight parallel circuit is connected in series with a circuit for a car stereo that can be modeled by a \(1.0\) resistor. What is the current being drawn from a 12 -volt battery when both the lights and the stereo are on? (Ans. \(\boldsymbol{I}=\mathbf{5 . 5} \mathbf{A}\).)
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