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Chapter 21: Problem 78

In a lightning bolt, a large amount of charge flows during a time of \(1.8 \times 10^{-3} \mathrm{s} .\) Assume that the bolt can be treated as a long, straight line of current. At a perpendicular distance of \(27 \mathrm{m}\) from the bolt, a magnetic field of \(8.0 \times 10^{-5} \mathrm{T}\) is measured. How much charge has flowed during the lightning bolt? Ignore the earth's magnetic field.

Short Answer

Expert verified
The charge that flowed during the lightning bolt is 1.944 C.

Step by step solution

01

Identify the formula for magnetic field due to a current

The magnetic field at a distance \( r \) from a long straight wire carrying current \( I \) is given by the formula \( B = \frac{\mu_0 I}{2\pi r} \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \text{ T m/A}) \), \( r \) is the distance from the wire, and \( I \) is the current.
02

Substitute known values to find current

We know that \( r = 27 \text{ m} \) and \( B = 8.0 \times 10^{-5} \ \text{T}\). Substituting these values into the formula gives: \[ 8.0 \times 10^{-5} = \frac{4\pi \times 10^{-7} \times I}{2\pi \times 27} \]. Simplifying this expression helps us solve for \( I \), the current.
03

Solve for the current, I

Simplifying the equation from Step 2: \[ I = \frac{8.0 \times 10^{-5} \times 2\pi \times 27}{4\pi \times 10^{-7}} \]. This simplifies to \[ I = \frac{8.0 \times 27 \times 10^{-5}}{2 \times 10^{-7}} \]. Calculating gives \( I = 1080 \text{ A} \).
04

Calculate the total charge

The total charge \( Q \) that flows can be calculated using the relationship \( Q = I \cdot t \), where \( t = 1.8 \times 10^{-3} \text{ s} \). So, \[ Q = 1080 \times 1.8 \times 10^{-3} \]. Performing the multiplication gives \( Q = 1.944 \text{ C} \).

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Key Concepts

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

Current
When we talk about current, we are essentially referring to the flow of electric charge through a conductor. In the context of a lightning bolt, this conductor is the path through the air that the lightning takes. Electric current is often measured in amperes, abbreviated as A. An ampere represents a flow of one coulomb of charge per second.

For a lightning bolt, the current can be gigantic, sometimes hundreds to thousands of amperes. This means an enormous amount of electric charge is moving through a very short period. Understanding this helps to appreciate the powerful effects and the intense energy release during a lightning strike.
  • Current is the rate of flow of electrical charge, denoted by the symbol \( I \).
  • Measured in amperes (A), with 1 ampere equal to 1 coulomb per second.
  • In the given example, the current during the lightning bolt was calculated to be 1080 A.
Charge Flow
Charge flow is simply how much electric charge moves through the conductor over time. Lightning bolts involve tremendous charge flow due to the quick and large transfer of charges from cloud to ground or between clouds. The amount of charge that flows during such an event is significant and is measured in coulombs.

The relationship between current and charge flow in physics is mathematical. It can be expressed as \( Q = I \cdot t \), where \( Q \) is the total charge, \( I \) is the current, and \( t \) is the time over which the current flows. This formula helps to calculate the magnitude of charge that moves鈥攁 critical piece of understanding for studying the impressive energies involved in a lightning strike.
  • The total charge flowing is denoted as \( Q \), measured in coulombs (C).
  • In our example, the charge flow during the 1.8 ms of the lightning bolt was \( 1.944 \) C.
Lightning Bolt
A lightning bolt is a massive natural electric discharge of static electricity. It usually occurs in a thunderstorm and is a dramatic illustration of electric and magnetic field interactions. Lightning strikes the ground or moves between clouds when there's a building of charge in the atmosphere, a sudden release is triggered, manifesting as a lightning strike.

During this event, the moving charges generate a strong magnetic field. This is why lightning bolts are associated with measurable magnetic effects. For instance, at a perpendicular distance from the lightning sheet, a magnetic field can be detected, influencing the environment around it.
  • Lightning is a natural high voltage discharge that can produce a temporary immense current.
  • It involves significant charge flow, capable of creating notable magnetic effects measurable far from the strike location.
  • In the provided example, the lightning bolt induced a magnetic field of \( 8.0 \times 10^{-5} \text{ T} \) at 27 m away.

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Most popular questions from this chapter

In New England, the horizontal component of the earth's magnetic field has a magnitude of \(1.6 \times 10^{-5} \mathrm{T} .\) An electron is shot vertically straight up from the ground with a speed of \(2.1 \times 10^{6} \mathrm{m} / \mathrm{s} .\) What is the magnitude of the acceleration caused by the magnetic force? Ignore the gravitational force acting on the electron.

A piece of copper wire has a resistance per unit length of \(5.90 \times 10^{-3} \Omega / \mathrm{m} .\) The wire is wound into a thin, flat coil of many turns that has a radius of \(0.140 \mathrm{m}\). The ends of the wire are connected to a \(12.0-\mathrm{V}\) battery. Find the magnetic field strength at the center of the coil.

A small compass is held horizontally, the center of its needle a distance of \(0.280 \mathrm{m}\) directly north of a long wire that is perpendicular to the earth's surface. When there is no current in the wire, the compass needle points due north, which is the direction of the horizontal component of the earth's magnetic field at that location. This component is parallel to the earth's surface. When the current in the wire is \(25.0 \mathrm{A}\), the needle points \(23.0^{\circ}\) east of north. (a) Does the current in the wire flow toward or away from the earth's surface? (b) What is the magnitude of the horizontal component of the earth's magnetic field at the location of the compass?

A long solenoid has a length of \(0.65 \mathrm{m}\) and contains 1400 turns of wire. There is a current of \(4.7 \mathrm{A}\) in the wire. What is the magnitude of the magnetic field within the solenoid?

An \(\alpha\) -particle has a charge of \(+2 e\) and a mass of \(6.64 \times 10^{-27} \mathrm{kg} .\) It is accelerated from rest through a potential difference that has a value of \(1.20 \times 10^{6} \mathrm{V}\) and then enters a uniform magnetic field whose magnitude is \(2.20 \mathrm{T}\). The \(\alpha\) -particle moves perpendicular to the magnetic field at all times. What is (a) the speed of the \(\alpha\) -particle, (b) the magnitude of the magnetic force on it, and (c) the radius of its circular path?
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