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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
38.88 C

Step by step solution

01

Identify the Given Values

First, we identify the values given in the problem. We have:- Distance from the bolt to the point where the magnetic field is measured, \( r = 27 \text{ m} \).- Magnetic field strength, \( B = 8.0 \times 10^{-5} \text{ T} \).- Time during which the charge flows, \( t = 1.8 \times 10^{-3} \text{ s} \).
02

Use the Magnetic Field Formula for a Current-Carrying Wire

The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying a current \( I \) is given by the formula:\[ B = \frac{\mu_0 I}{2\pi r} \]where \( \mu_0 \) (the permeability of free space) is \( 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \).
03

Solve for Current \( I \)

Rearranging the formula to solve for \( I \), we have:\[ I = \frac{2\pi r B}{\mu_0} \]Substitute the given values into this equation:\[ I = \frac{2\pi (27) (8.0 \times 10^{-5})}{4\pi \times 10^{-7}} \]
04

Calculate the Current \( I \)

Simplify the equation:\[ I = \frac{2 \times 27 \times 8.0 \times 10^{-5}}{4 \times 10^{-7}} \]Calculate:\[ I \approx \frac{864 \times 10^{-5}}{4 \times 10^{-7}} \]\[ I \approx \frac{864}{4} \times 10^{2} \]\[ I \approx 216 \times 10^{2} \] \[ I \approx 2.16 \times 10^{4} \text{ A} \]
05

Calculate Total Charge \( Q \)

The charge \( Q \) that flows through the lightning bolt is given by the equation:\[ Q = I \times t \]Substitute the known values:\[ Q = 2.16 \times 10^{4} \times 1.8 \times 10^{-3} \]\[ Q = 3.888 \times 10^{1} \text{ C} \]
06

Final Result

The final charge that flowed during the lightning bolt is approximately:\[ Q \approx 38.88 \text{ C} \]

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Key Concepts

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

Understanding Lightning Bolts
Lightning bolts are fascinating and powerful natural phenomena that occur during thunderstorms. They are essentially massive electric discharges in the atmosphere. When a lightning bolt strikes, it releases a huge amount of energy within a very short period. In this exercise, it's modeled as a long, straight line of current to simplify understanding.

  • Charge Flow: The charge flow in a lightning bolt can be immense, which is why they have such destructive power. In our example, we calculated the total charge that flowed to be approximately 38.88 Coulombs.
  • Current and Time: We learned that the time for which the current flowed was only 1.8 milliseconds. Yet, the current was as high as 21,600 Amperes. This illustrates the burst-like nature of lightning strikes.
Lightning is a spectacular reminder of nature's power and the principles of electromagnetism at play.
What Happens in a Current-Carrying Wire
In physics, when we talk about current-carrying wires, we refer to wires through which electric current flows. The concept becomes crucial in understanding how different physical interactions occur around the wire.
  • For a long straight wire, the magnetic field produced at any point around it is directly related to the amount of current passing through the wire.
  • The formula to calculate the magnetic field due to a current-carrying wire is \[ B = \frac{\mu_0 I}{2\pi r} \], where \( B \) is the magnetic field, \( I \) is the current, \( r \) is the distance from the wire, and \( \mu_0 \) is the permeability of free space.
  • Rearranging this helps in calculating the amount of current flowing, as seen in the lightning bolt exercise.
This understanding helps bridge our knowledge to how electric circuits operate in everyday life.
Exploring Magnetic Fields
Magnetic fields are invisible regions around a magnet or current-carrying wire where magnetic forces can be felt. They play a crucial role in electromagnetism, influencing how charges move and interact.
  • Magnetic fields are represented by field lines; the denser the field lines, the stronger the magnetic field.
  • In our example, measuring a magnetic field of \(8.0 \times 10^{-5} \text{ T}\) at a point 27 meters from the lightning bolt indicates the presence of a substantial current.
  • By knowing the magnetic field strength at a particular point, we can back-calculate to determine the current flowing through a conductor using the appropriate formulas.
These concepts are used widely in technologies ranging from MRI machines to the design of electric motors and transformers.

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Most popular questions from this chapter

A particle has a charge of \(q=+5.60 \mu \mathrm{C}\) and is located at the coordinate origin. As the drawing shows, an electric field of \(E_{x}=+245 \mathrm{N} / \mathrm{C}\) exists along the \(+x\) axis. A magnetic field also exists, and its \(x\) and \(y\) components are \(B_{x}=+1.80 \mathrm{T}\) and \(B_{y}=+1.40 \mathrm{T} .\) Calculate the force (magnitude and direction) exerted on the particle by each of the three fields when it is (a) stationary, (b) moving along the \(+x\) axis at a speed of \(375 \mathrm{m} / \mathrm{s},\) and (c) moving along the \(+z\) axis at a speed of \(375 \mathrm{m} / \mathrm{s}\).

A horizontal wire of length \(0.53 \mathrm{m},\) carrying a current of \(7.5 \mathrm{A},\) is placed in a uniform external magnetic field. When the wire is horizontal, it experiences no magnetic force. When the wire is tilted upward at an angle of \(19^{\circ},\) it experiences a magnetic force of \(4.4 \times 10^{-3} \mathrm{N} .\) Determine the magnitude of the external magnetic field.

A charged particle enters a uniform magnetic field and follows the circular path shown in the drawing. (a) Is the particle positively or negatively charged? Why? (b) The particle's speed is \(140 \mathrm{m} / \mathrm{s},\) the magnitude of the magnetic field is \(0.48 \mathrm{T}\), and the radius of the path is \(960 \mathrm{m}\). Determine the mass of the particle, given that its charge has a magnitude of \(8.2 \times 10^{-4} \mathrm{C}\).

A 125-turn rectangular coil of wire is hung from one arm of a balance, as the figure shows. With the magnetic field \(\overrightarrow{\mathbf{B}}\) turned off, an object of mass \(M\) is added to the pan on the other arm to balance the weight of the coil. Concepts: (i) In a balanced, or equilibrium, condition the device has no angular acceleration. What does this imply about the net torque acting on the device? (ii) What is torque? (iii) In calculating the torques acting on an object in equilibrium, where do you locate the axis of rotation? Calculations: When a constant 0.200-T magnetic field is turned on and there is a current of \(8.50 \mathrm{A}\) in the coil, how much additional mass \(m\) must be added to regain the balance?

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?
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