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How long does it take electrons to get from a car battery to the starting motor? Assume the current is \(300 \mathrm{~A}\) and the electrons travel through a copper wire with cross-sectional area \(0.21 \mathrm{~cm}^{2}\) and length \(0.85 \mathrm{~m} .\) The number of charge carriers per unit volume is \(8.49 \times 10^{28} \mathrm{~m}^{-3}\).

Short Answer

Expert verified
The electrons take approximately 30.8 minutes to get from the battery to the motor.

Step by step solution

01

Understand the formula

To find the time it takes electrons to travel a certain distance, we use the drift velocity formula.The drift velocity \( v_d \) can be calculated using the formula: \[ v_d = \frac{I}{n \cdot A \cdot e} \]where \( I \) is the current, \( n \) is the number of charge carriers per unit volume, \( A \) is the cross-sectional area of the wire, and \( e \) is the elementary charge (approximately \(1.6 \times 10^{-19} \mathrm{~C}\)).
02

Calculate drift velocity

Given, \( I = 300 \mathrm{~A} \), \( n = 8.49 \times 10^{28} \mathrm{~m}^{-3} \), \( A = 0.21 \mathrm{~cm}^2 = 0.21 \times 10^{-4} \mathrm{~m}^2 \).Plug these values into the drift velocity formula:\[ v_d = \frac{300}{8.49 \times 10^{28} \times 0.21 \times 10^{-4} \times 1.6 \times 10^{-19}} \]Calculate this to get \( v_d \approx 4.6 \times 10^{-4} \mathrm{~m/s} \).
03

Calculate the time taken

Using the drift velocity, calculate the time taken \( t \) to travel the distance \( L \) (length of wire given as \( 0.85 \mathrm{~m} \)). The formula to find time is:\[ t = \frac{L}{v_d} \]Substitute \( L = 0.85 \mathrm{~m} \) and \( v_d = 4.6 \times 10^{-4} \mathrm{~m/s} \) into the equation: \[ t = \frac{0.85}{4.6 \times 10^{-4}} \]Calculate this to get \( t \approx 1848 \mathrm{~s} \).
04

Convert time to minutes

To convert the time from seconds to minutes, divide by 60 (since there are 60 seconds in a minute): \[ t = \frac{1848}{60} \]Calculate this to get \( t \approx 30.8 \mathrm{~minutes} \).

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Key Concepts

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

Current Electricity
Current electricity refers to the flow of electric charge through a conductor. In electrical circuits, this flow is made up primarily of the individual movement of electrons. Understanding the core principles of current electricity is important because it forms the foundation of how most modern electrical devices operate.

An electrical circuit consists of a closed loop, through which electrons move, driven by a potential difference, or voltage. This movement creates an electric current. The two main types of current are direct current (DC), where charge flows in one direction, and alternating current (AC), where the flow of charge periodically reverses direction.
  • It is the backbone of both simple electronic devices, like a flashlight, and large systems, like power grids.
  • Established by the movement of electrons across conductive materials.
This movement is proportional to the voltage applied and inversely proportional to the resistance in the circuit, as described by Ohm's law: \( V = IR \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.
Electron Flow
Electron flow refers to the motion of electrons through a conductor, such as a copper wire. This movement is what constitutes an electric current. Electrons carry a negative charge, and in metallic conductors, they are usually free to move between atoms.
  • Electrons flow from areas of negatively charged potential to positively charged areas.
  • In conventional circuits, this flow is from the negative terminal to the positive terminal of a power source, like a battery.
The concept of drift velocity helps describe how fast electrons travel through a conductor under the influence of an electric field. Despite the large number of charge carriers in a wire, individual electrons move relatively slowly. It is the composition of these small displacements that results in an observable flow, thanks to the immense number of electrons present in a conductor.
Electric Current
Electric current is the rate at which charge flows through a surface or conductor. Often measured in amperes (A), it is one of the fundamental measurements in electrical engineering.

There are two main types of electric currents:
  • Direct Current (DC) – where charge flows in a single direction. Commonly found in batteries and electronic devices.
  • Alternating Current (AC) – where the direction of charge flow periodically changes. It is the type of current usually delivered by power plants and used in household appliances.
The amount of current that a wire can carry safely depends on its cross-sectional area and material. Too much current can cause overheating and damage. Therefore, understanding resistance and how it works with electric current is critical for safe electrical design.
Copper Wire Conductivity
Copper is one of the most widely used conductors in electrical wiring due to its excellent conductive properties. The conductivity of copper makes it a reliable choice for carrying electric current efficiently over distances.

Key characteristics include:
  • High electrical conductivity: Copper provides low electrical resistance, reducing energy loss as heat.
  • Durability and flexibility: These physical properties make copper suitable for being bent and shaped without breaking, which is necessary for wiring.
  • Abundant availability: Copper is more readily available and economical than metals like silver, which has slightly better conductivity.
In the context of drift velocity, a copper wire's conductivity directly influences the ease and speed at which electrons can move through it. The drift velocity calculation shows that despite the slow speed of individual electrons, the overall effect of their mass movement is rapid signal transmission. This makes copper an excellent choice for electrical and telecommunication systems.

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Most popular questions from this chapter

Earth's lower atmosphere contains negative and positive ions that are produced by radioactive elements in the soil and cosmic rays from space. In a certain region, the atmospheric electric field strength is \(120 \mathrm{~V} / \mathrm{m}\) and the field is directed vertically down. This field causes singly charged positive ions, at a density of \(620 \mathrm{~cm}^{-3}\), to drift downward and singly charged negative ions, at a density of \(550 \mathrm{~cm}^{-3}\), to drift upward (Fig. \(\left.26-27\right)\). The measured conductivity of the air in that region is \(2.70 \times 10^{-14}\) \((\Omega \cdot \mathrm{m})^{-1}\). Calculate (a) the magnitude of the current density and (b) the ion drift speed, assumed to be the same for positive and negative ions.

The current density in a wire is uniform and has magnitude \(2.0 \times 10^{6} \mathrm{~A} / \mathrm{m}^{2}\), the wire's length is \(5.0 \mathrm{~m}\), and the density of conduction electrons is \(8.49 \times 10^{28} \mathrm{~m}^{-3}\). How long does an electron take (on the average) to travel the length of the wire?

A steel trolley-car rail has a cross-sectional area of \(56.0 \mathrm{~cm}^{2}\). What is the resistance of \(10.0 \mathrm{~km}\) of rail? The resistivity of the steel is \(3.00 \times 10^{-7} \Omega \cdot \mathrm{m}\)

A \(500 \mathrm{~W}\) heating unit is designed to operate with an applied potential difference of \(115 \mathrm{~V} .\) (a) By what percentage will its heat output drop if the applied potential difference drops to \(110 \mathrm{~V} ?\) Assume no change in resistance. (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than that calculated in (a)?

Kiting during a storm. The legend that Benjamin Franklin flew a kite as a storm approached is only a legend-he was neither stupid nor suicidal. Suppose a kite string of radius \(2.00\) \(\mathrm{mm}\) extends directly upward by \(0.800 \mathrm{~km}\) and is coated with a \(0.500\) \(\mathrm{mm}\) layer of water having resistivity \(150 \Omega \cdot \mathrm{m}\). If the potential difference between the two ends of the string is \(160 \mathrm{MV}\), what is the current through the water layer? The danger is not this current but the chance that the string draws a lightning strike, which can have a current as large as \(500000 \mathrm{~A}\) (way beyond just being lethal).

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