Physics Lesson 16.18.1 - Gauss Law for Magnetic Field

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Welcome to our Physics lesson on Gauss Law for Magnetic Field, this is the first lesson of our suite of physics lessons covering the topic of Maxwell Equations, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Gauss Law for Magnetic Field

As explained in tutorial 16.1 "Introduction to Magnetism", all magnets have two poles: one north and one south. The magnetic field lines are closes; they start from the north-pole and end at the south-pole of the same magnet. As and evidence for this, we can consider the shape the small iron filings poured on a paper sheet take when a magnet is placed below the sheet.

Physics Tutorials: This image provides visual information for the physics tutorial Maxwell Equations

We have also provided (in the same tutorial) the definition of magnetic dipole, which is the smallest possible magnet made from a proton-electron pair. This means the magnetic monopoles do not exist.

Physics Tutorials: This image provides visual information for the physics tutorial Maxwell Equations

The Gauss Law for the magnetic field implies that magnetic monopoles do not exist. Mathematically, this law means that the net magnetic flux Φm through any closed Gaussian surface is zero. The Gauss Law formula for magnetic field is

ΦM = B ∙ dA = 0

This outcome is different from the Gauss Law in electric fields. Recall that in electric fields, we have

Φe = E ∙ dA = Q/ε0

Unlike in electric fields in which the net electric flux through a closed Gaussian surface is proportional to the charge, in magnetic field the net magnetic flux through a Gaussian surface is zero. This is because the net magnetic charge in all magnets is zero because the charge is balanced; in magnets, we simply have a certain regular alignment of dipoles but the net charge is zero.

The figure below gives a clearer idea on this point.

Physics Tutorials: This image provides visual information for the physics tutorial Maxwell Equations

As you see from the figure, there are two lines entering the loop and two lines leaving the loop. Therefore, the net flux flowing through the Amperian loop is zero.

Example 1

In which of the points shown in the figure

  1. the magnetic field is the greatest?
  2. the magnetic flux is the greatest?
Physics Tutorials: This image provides visual information for the physics tutorial Maxwell Equations

Solution 1

  1. The magnetic field is strongest near the poles as the magnetic field lines have a higher density in those regions. In addition, the magnetic field decreases with the increase in distance from the magnet. Therefore, the order of magnetic fields in the four given points from the smallest to the largest is
    Bd < Bc < Bb < B a
    Hence, the magnetic field is the strongest at the point A.
  2. As for the magnetic flux, it is zero everywhere as the number of magnetic field lines entering from up at any closed Amperian loop around the given points is equal to the number of lines leaving the loop, regardless the numbers vary according the corresponding magnetic field strengths. For example, we may have 10 lines entering a closed loop around the point A and 10 lines leaving the loop, and for example 7 lines entering the same closed loop around the point B and 7 lines leaving the loop. In both cases the total flux is zero, regardless the magnetic field is different.

You have reached the end of Physics lesson 16.18.1 Gauss Law for Magnetic Field. There are 5 lessons in this physics tutorial covering Maxwell Equations, you can access all the lessons from this tutorial below.

More Maxwell Equations Lessons and Learning Resources

Magnetism Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
16.18Maxwell Equations
Lesson IDPhysics Lesson TitleLessonVideo
16.18.1Gauss Law for Magnetic Field
16.18.2Induced Magnetic Fields
16.18.3Displacement Current
16.18.4How to Find the Induced Magnetic Field?
16.18.5Maxwell Equations Explained

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