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Chapter 27: Problem 8

A certain car battery with a \(12.0 \mathrm{~V}\) emf has an initial charge of \(120 \mathrm{~A} \cdot \mathrm{h}\). Assuming that the potential across the terminals stays constant until the battery is completely discharged, for how many hours can it deliver energy at the rate of \(100 \mathrm{~W} ?\)

Short Answer

Expert verified
The battery can deliver energy at 100 W for approximately 14.4 hours.

Step by step solution

01

Convert Power to Current

First, we need to find the current that the battery delivers. We can use the formula for electric power: \[ P = V imes I \]where \(P\) is the power (\(100\, \mathrm{W}\)), \(V\) is the voltage (\(12.0\, \mathrm{V}\)), and \(I\) is the current. Solving for \(I\), we have:\[ I = \frac{P}{V} = \frac{100}{12.0} = 8.33 \mathrm{~A} \].
02

Find Time to Discharge

Next, we find the time it takes to discharge the battery. The initial charge of the battery is given as \(120 \mathrm{~A} \cdot \mathrm{h}\). The battery discharges at the rate of \(8.33 \mathrm{~A}\). To find the time \(t\) in hours, we use the formula for charge \(Q = I \times t\). Solving for time:\[ t = \frac{Q}{I} = \frac{120}{8.33} \approx 14.4 \mathrm{~h} \].

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Key Concepts

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

Battery Emf
Electromotive force (Emf) is a crucial concept in understanding how batteries work. It refers to the potential difference between the two terminals of a battery when no current is flowing. In simpler terms, it's the maximum "push" a battery can give to electrons to generate electricity. A battery with a higher Emf can deliver more power to a device.
  1. The Emf describes the capability of a battery to supply energy before any load is connected.
  2. In a "12.0 V" battery, 12 volts represent the Emf.
  3. This Emf remains relatively stable until the battery approaches full discharge.
Remember that Emf doesn't represent the energy the battery can permanently supply; rather, it highlights the voltage throughout its operation.
Current Calculation
Calculating current involves understanding how much electric charge flows per second in a circuit. When we know the power, in watts, and the voltage of a system, calculating the current is straightforward using the formula: \[ P = V \times I \] Where:
  • P is Power in watts.
  • V is Voltage in volts.
  • I is Current in amperes.
In the context of the exercise, we needed the current to find out how long the battery could supply a particular power output of 100 W. By rearranging the formula to \( I = \frac{P}{V} \), the current \( I \) for a 12.0V battery delivering 100 W came out to be approximately 8.33 A. This calculation is pivotal because knowing the current helps us determine battery life and energy efficiency for devices powered by that battery.
Charge and Discharge Time
Knowing how long a battery can last is essential for planning its use in various devices. Battery capacity is often expressed in ampere-hours (A路h), which signifies the total charge a battery can hold. To determine how long a battery can sustain a specific load, the key formula is:\[ t = \frac{Q}{I} \]
  • t is the time, in hours, the battery can supply the load.
  • Q is the charge, typically given in ampere-hours.
  • I is the current, in amperes, flowing from the battery.
For the car battery in question, the initial charge is 120 A路h. When delivering a current of 8.33 A, we calculate \( t = \frac{120}{8.33} \approx 14.4 \) hours. This means that under a constant power draw of 100 W, the battery would take around 14.4 hours to deplete entirely. This concept is crucial for designing battery-operated systems to ensure they operate long enough between charges.

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Most popular questions from this chapter

The starting motor of a car is turning too slowly, and the mechanic has to decide whether to replace the motor, the cable, or the battery. The car's manual says that the \(12 \mathrm{~V}\) battery should have no more than \(0.020 \Omega\) internal resistance, the motor no more than \(0.200 \Omega\) resistance, and the cable no more than \(0.040 \Omega\) resistance. The mechanic turns on the motor and measures \(11.4 \mathrm{~V}\) across the battery, \(3.0 \mathrm{~V}\) across the cable, and a current of \(50 \mathrm{~A}\). Which part is defective?

A controller on an electronic arcade game consists of a variable resistor connected across the plates of a \(0.220 \mu \mathrm{F}\) capacitor. The capacitor is charged to \(5.00 \mathrm{~V}\), then discharged through the resistor. The time for the potential difference across the plates to decrease to \(0.800 \mathrm{~V}\) is measured by a clock inside the game. If the range of discharge times that can be handled effectively is from \(10.0 \mu \mathrm{s}\) to \(6.00 \mathrm{~ms}\), what should be the (a) lower value and (b) higher value of the resistance range of the resistor?

An automobile gasoline gauge is shown schematically in Fig. \(27-74\). The indicator (on the dashboard) has a resistance of \(10 \Omega\). The tank unit is a float connected to a variable resistor whose resistance varies linearly with the volume of gasoline. The resistance is \(140 \Omega\) when the tank is empty and \(20 \Omega\) when the tank is full. Find the current in the circuit when the tank is (a) empty, (b) half-full, and (c) full. Treat the battery as ideal.

A solar cell generates a potential difference of \(0.10 \mathrm{~V}\) when a \(500 \Omega\) resistor is connected across it, and a potential difference of \(0.15 \mathrm{~V}\) when a \(1000 \Omega\) resistor is substituted. What are the (a) internal resistance and (b) emf of the solar cell? (c) The area of the cell is \(5.0 \mathrm{~cm}^{2}\), and the rate per unit area at which it receives energy from light is \(2.0 \mathrm{~mW} / \mathrm{cm}^{2}\). What is the efficiency of the cell for converting light energy to thermal energy in the \(1000 \Omega\) external resistor?

When resistors 1 and 2 are connected in series, the equivalent resistance is \(16.0 \Omega\). When they are connected in parallel, the equivalent resistance is \(3.0 \Omega\). What are (a) the smaller resistance and (b) the larger resistance of these two resistors?
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