Tru Physics

Chapter 18: The Magnetic Field of a Moving Charge

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18.1 Introduction

In this chapter, we will discuss the magnetic field produced by moving charges, such as electrons and ions, and how it affects the motion of other charged particles. Understanding the magnetic field of a moving charge is essential for various applications, including the design of particle accelerators and the analysis of plasma behavior in fusion reactors.

Moving charges, even in a household bulb like this, will produce magnetic fields. With precise equipment, these field can be measured and verified.

18.2 Biot-Savart Law

The Biot-Savart law describes the magnetic field generated by a current-carrying conductor. It states that the magnetic field d\vec{B} produced by a small segment of a current-carrying wire at a point P is proportional to the current I, the length of the segment d\vec{L}, and the sine of the angle \theta between d\vec{L} and the unit vector \hat{r} connecting the segment to point P, divided by the square of the distance |\vec{r}|^2:

d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I d\vec{L} \times \hat{r}}{|\vec{r}|^2}

where \mu_0 is the permeability of free space.

18.3 Magnetic Field Due to a Moving Charge

When a single charged particle, such as an electron, moves with velocity \vec{v}, it generates a magnetic field. To find the magnetic field \vec{B} at a point P due to a moving charge q, we can consider the charge as a current I in an infinitesimally small wire segment d\vec{L}, where I = \frac{dq}{dt} and d\vec{L} = \vec{v}dt:

\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{q\vec{v} \times \vec{r}}{|\vec{r}|^3}

18.4 Motion of Charged Particles in a Magnetic Field

A charged particle moving in a magnetic field experiences a force called the Lorentz force, given by:

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

where q is the charge of the particle, \vec{E} is the electric field, \vec{v} is the velocity of the particle, and \vec{B} is the magnetic field.

If the particle is moving perpendicular to the magnetic field, its path will be a circle with a radius r, given by:

r = \dfrac{mv}{|q|\vec{B}}

where m is the mass of the particle.

If the particle is moving at an angle relative to the magnetic field, its path will be a helix, with its pitch and radius determined by the velocity components parallel and perpendicular to the magnetic field.

18.5 Applications of the Magnetic Field of Moving Charges

The magnetic field produced by moving charges plays a vital role in various scientific and technological applications, such as:

  • Particle accelerators: Charged particles are accelerated to high speeds using electric and magnetic fields, enabling physicists to study fundamental particles and forces.
  • Magnetic confinement fusion: In fusion reactors, magnetic fields are used to confine high-temperature plasmas, allowing for nuclear fusion to occur and produce energy.
  • Mass spectrometry: Magnetic fields are employed to separate ions based on their mass-to-charge ratio, enabling the analysis of the composition of samples in various fields, such as chemistry, biology, and environmental science.

Chapter Summary

In this chapter, we discussed the magnetic field produced by moving charges, the motion of charged particles in a magnetic field, and the importance of understanding the magnetic field of moving charges in various applications. This knowledge is crucial for the design and analysis of systems that involve the interaction of charged particles and magnetic fields.

Continue to Chapter 19: Ampere’s Law

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