Tru Physics

Magnetic Field

—

by

Tru Physics
in

Introduction

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength), so it is a vector field.

Magnetic Force on a Moving Charge

The force exerted on a charge q moving with velocity \vec{v} in a magnetic field \vec{B} is given by the Lorentz Force Law:

\vec{F} = q\vec{v} \times \vec{B}

The direction of the force is perpendicular to both the velocity of the charge and the magnetic field, following the right-hand rule.

Magnetic Field due to a Current

Biot-Savart Law

The magnetic field \vec{B} at a point due to a small element of a wire carrying a current I is given by the Biot-Savart Law:

d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{Id\vec{l} \times \hat{r}}{r^2}

where d\vec{l} is a small length element of the wire, \hat{r} is the unit vector from the current element to the point r in space, and \mu_0 is the permeability of free space.

Ampere’s Law

Ampere’s Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop:

\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}

where d\vec{l} is an infinitesimal vector element of a closed loop, and I_{\text{enc}} is the current enclosed by the loop.

Magnetic Field due to a Solenoid

The magnetic field inside a long solenoid carrying a current I is uniform and parallel to the axis of the solenoid. It is given by:

\vec{B} = \mu_0 n I

where n is the number of turns per unit length of the solenoid.

Magnetic Field due to a Toroid

The magnetic field inside a toroid is uniform and circular. It is given by:

\vec{B} = \dfrac{\mu_0 NI}{2\pi r}

where N is the total number of turns, I is the current, and r is the radius from the center of the toroid.

Magnetic Flux and Faraday’s Law of Induction

The magnetic flux \Phi_B through a surface is given by:

\Phi_B = \int \vec{B} \cdot d\vec{A}

where d\vec{A} is an infinitesimal element of the surface.

Faraday’s Law of Induction states that the electromotive force (EMF) \varepsilon induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop:

\varepsilon = -\dfrac{d\Phi_B}{dt}

Maxwell’s Equations

Magnetic fields are one of the key elements in Maxwell’s equations, the four fundamental equations that describe classical electrodynamics:

  1. Gauss’s law for electricity: \nabla \cdot \vec{E} = \dfrac{\rho}{\varepsilon_0}
  2. Gauss’s law for magnetism: \nabla \cdot \vec{B} = 0
  3. Faraday’s law of induction: \nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}
  4. Ampere-Maxwell law: \nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\dfrac{\partial \vec{E}}{\partial t}

Advanced Topics: Quantum Mechanics and Magnetism

In quantum mechanics, the interaction of particles with a magnetic field is described by the addition of a magnetic term to the Hamiltonian operator in the Schrödinger equation:

\hat{H} = \dfrac{(\hat{p}-q\vec{A})^2}{2m} + q\phi

where \hat{p} is the momentum operator, q is the charge of the particle, \vec{A} is the magnetic vector potential, and \phi is the electric scalar potential.

In quantum field theory, the magnetic field is described as a result of exchange of virtual photons, the force-carrying particles of the electromagnetic force. The interaction of magnetic moments with the magnetic field is also a topic of interest in quantum electrodynamics (QED).

Magnetic Monopoles

In classical electrodynamics, magnetic monopoles, which are isolated north or south magnetic poles, do not exist. However, in some quantum mechanical theories and in some speculative unified theories, magnetic monopoles are predicted to exist. These theories often involve modifications to Maxwell’s equations or the introduction of additional fields. Despite extensive searches, as of the time of writing, no magnetic monopole has been experimentally detected (and not for lack of effort).

The study of magnetic fields and their effects is a vast topic that is central to many areas of physics, including classical electrodynamics, quantum mechanics, quantum field theory, condensed matter physics, and astrophysics.

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Want to study more? Visit our Index here!

Comments

One response to “Magnetic Field”

  1. Ampere’s Law – Tru Physics

    […] is the line integral of the magnetic field around a closed loop, is the permeability of free space, and is the electric current passing […]

    Reply

Have something to add? Leave a comment!