What Magnetic Field Strength Is Needed To Exert A Force Of $1.0 \times 10^{-15} \, \text{N}$ On An Electron Traveling At $2.0 \times 10^7 \, \text{m/s}$?

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Introduction

The force exerted on a charged particle by a magnetic field is a fundamental concept in physics, governed by the Lorentz force equation. In this article, we will explore the relationship between the magnetic field strength, the velocity of the charged particle, and the force exerted on it. Specifically, we will determine the magnetic field strength required to exert a force of $1.0 \times 10^{-15} , \text{N}$ on an electron traveling at $2.0 \times 10^7 , \text{m/s}$.

The Lorentz Force Equation

The Lorentz force equation is a fundamental concept in physics that describes the force exerted on a charged particle by an electric and magnetic field. The equation is given by:

F=q(E+v×B)\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})

where F\mathbf{F} is the force exerted on the charged particle, qq is the charge of the particle, E\mathbf{E} is the electric field, v\mathbf{v} is the velocity of the particle, and B\mathbf{B} is the magnetic field.

Magnetic Field and Force Relationship

In the absence of an electric field, the Lorentz force equation reduces to:

F=q(v×B)\mathbf{F} = q(\mathbf{v} \times \mathbf{B})

This equation shows that the force exerted on a charged particle by a magnetic field is perpendicular to both the velocity of the particle and the magnetic field. The magnitude of the force is given by:

F=qvBsin⁡θF = qvB\sin\theta

where θ\theta is the angle between the velocity and the magnetic field.

Calculating the Magnetic Field Strength

To calculate the magnetic field strength required to exert a force of $1.0 \times 10^{-15} , \text{N}$ on an electron traveling at $2.0 \times 10^7 , \text{m/s}$, we can use the equation:

F=qvBsin⁡θF = qvB\sin\theta

We know that the charge of an electron is $1.6 \times 10^{-19} , \text{C}$, the velocity of the electron is $2.0 \times 10^7 , \text{m/s}$, and the force exerted is $1.0 \times 10^{-15} , \text{N}$. We can assume that the angle between the velocity and the magnetic field is $90^\circ$, so $\sin\theta = 1$.

Rearranging the equation to solve for the magnetic field strength, we get:

B=FqvB = \frac{F}{qv}

Substituting the values, we get:

B=1.0×10−15 N(1.6×10−19 C)(2.0×107 m/s)B = \frac{1.0 \times 10^{-15} \, \text{N}}{(1.6 \times 10^{-19} \, \text{C})(2.0 \times 10^7 \, \text{m/s})}

Simplifying the expression, we get:

B=3.125×10−4 TB = 3.125 \times 10^{-4} \, \text{T}

Conclusion

In this article, we have determined the magnetic field strength required to exert a force of $1.0 \times 10^{-15} , \text{N}$ on an electron traveling at $2.0 \times 10^7 , \text{m/s}$. We have used the Lorentz force equation and the equation for the force exerted by a magnetic field to calculate the magnetic field strength. The result shows that a magnetic field strength of $3.125 \times 10^{-4} , \text{T}$ is required to exert the specified force on the electron.

References

  • [1] Feynman, R. P. (1963). The Feynman Lectures on Physics. Addison-Wesley.
  • [2] Griffiths, D. J. (2013). Introduction to Electrodynamics. Pearson Education.
  • [3] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.

Further Reading

  • [1] Lorentz, H. A. (1892). The Theory of Electrons. B.G. Teubner.
  • [2] Maxwell, J. C. (1864). A Dynamical Theory of the Electromagnetic Field. Royal Society.
  • [3] Thomson, J. J. (1897). Electricity and Matter. Charles Scribner's Sons.

Related Topics

  • [1] Electric Field and Force
  • [2] Magnetic Field and Force
  • [3] Lorentz Force Equation

Tags

  • [1] Magnetic Field
  • [2] Force
  • [3] Lorentz Force Equation
  • [4] Electron
  • [5] Velocity
  • [6] Charge
  • [7] Electric Field
  • [8] Physics

Q: What is the Lorentz force equation, and how is it used to calculate the force exerted on a charged particle by a magnetic field?

A: The Lorentz force equation is a fundamental concept in physics that describes the force exerted on a charged particle by an electric and magnetic field. The equation is given by:

F=q(E+v×B)\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})

where F\mathbf{F} is the force exerted on the charged particle, qq is the charge of the particle, E\mathbf{E} is the electric field, v\mathbf{v} is the velocity of the particle, and B\mathbf{B} is the magnetic field.

Q: How does the Lorentz force equation relate to the force exerted by a magnetic field?

A: In the absence of an electric field, the Lorentz force equation reduces to:

F=q(v×B)\mathbf{F} = q(\mathbf{v} \times \mathbf{B})

This equation shows that the force exerted on a charged particle by a magnetic field is perpendicular to both the velocity of the particle and the magnetic field.

Q: What is the equation for the force exerted by a magnetic field, and how is it used to calculate the magnetic field strength?

A: The equation for the force exerted by a magnetic field is given by:

F=qvBsin⁡θF = qvB\sin\theta

where θ\theta is the angle between the velocity and the magnetic field.

Q: How do you calculate the magnetic field strength required to exert a force of $1.0 \times 10^{-15} , \text{N}$ on an electron traveling at $2.0 \times 10^7 , \text{m/s}$?

A: To calculate the magnetic field strength required to exert a force of $1.0 \times 10^{-15} , \text{N}$ on an electron traveling at $2.0 \times 10^7 , \text{m/s}$, we can use the equation:

B=FqvB = \frac{F}{qv}

Substituting the values, we get:

B=1.0×10−15 N(1.6×10−19 C)(2.0×107 m/s)B = \frac{1.0 \times 10^{-15} \, \text{N}}{(1.6 \times 10^{-19} \, \text{C})(2.0 \times 10^7 \, \text{m/s})}

Simplifying the expression, we get:

B=3.125×10−4 TB = 3.125 \times 10^{-4} \, \text{T}

Q: What are some common applications of the Lorentz force equation and the force exerted by a magnetic field?

A: The Lorentz force equation and the force exerted by a magnetic field have many common applications in physics and engineering, including:

  • Particle accelerators
  • Magnetic resonance imaging (MRI)
  • Electric motors
  • Generators
  • Magnetic levitation (maglev) trains

Q: What are some common misconceptions about the Lorentz force equation and the force exerted by a magnetic field?

A: Some common misconceptions about the Lorentz force equation and the force exerted by a magnetic field include:

  • The force exerted by a magnetic field is always perpendicular to the velocity of the particle.
  • The force exerted by a magnetic field is always proportional to the charge of the particle.
  • The Lorentz force equation only applies to charged particles.

Q: How can I learn more about the Lorentz force equation and the force exerted by a magnetic field?

A: There are many resources available to learn more about the Lorentz force equation and the force exerted by a magnetic field, including:

  • Textbooks on electromagnetism and physics
  • Online courses and tutorials
  • Research papers and articles
  • Online forums and communities

Q: What are some common problems that can arise when using the Lorentz force equation and the force exerted by a magnetic field?

A: Some common problems that can arise when using the Lorentz force equation and the force exerted by a magnetic field include:

  • Incorrectly applying the Lorentz force equation to a given problem
  • Failing to account for the electric field in the Lorentz force equation
  • Using incorrect values for the charge, velocity, and magnetic field strength.

Q: How can I troubleshoot common problems that arise when using the Lorentz force equation and the force exerted by a magnetic field?

A: To troubleshoot common problems that arise when using the Lorentz force equation and the force exerted by a magnetic field, you can:

  • Double-check your calculations and equations
  • Verify that you have correctly applied the Lorentz force equation to the problem
  • Consult with a colleague or expert in the field
  • Use online resources and tutorials to learn more about the Lorentz force equation and the force exerted by a magnetic field.