Maxwell’s Equations – Integral form of Gauss’s Law for magnetic fields (part 3)





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This is Part three of my eight part series on Maxwell Equations

My last two articles namely

  1. Maxwell’s Equations – Integral form of Gauss’s Law (Part 1)
  2. Maxwell’s Equations – Differential form of Gauss’s Law (Part 2)

I discussed about both integral and differential form of Gauss’s Law for electric fields. In this article I’m going to discuss about second Maxwell’s equation which is about magnetic fields. So this article would be about Integral form of Gauss’s Law for magnetic fields.

Integral form of Gauss’s Law for magnetic field is written as

$\oint\limits_S {\overrightarrow B  \cdot \widehat nda}  = 0$

Here in above equation left side of is a mathematical description of the flux of a vector field through a closed surface. Here as you know Gauss’s law refers to magnetic flux which is the number of magnetic field lines passing through a closed surface S. The right side of the equation is zero.
Gauss’s Law for magnetic field states that

The total magnetic flux passing through any closed surface is zero

Surface mentioned here can be real or imaginary and can be of any shape and size. Here it is mentioned that flux through any closed surface must be zero but this does not mean that magnetic field lies does not pass through the surface it means that when an magnetic field line enters the surface there must be a magnetic field line leaving the volume enclosed by the surface under consideration.
Gauss’s Law of magnetic field arises directly as a result of lack existence of magnetic mono-poles in nature. So isolated magnetic poles simply does not exist. Every magnetic north pole is accompanied by a magnetic south pole. Thus the right side of Gauss’s law for magnetic fields is identically zero.
Now I’ll give a brief explanation of the meaning of each and every symbol used in Integral form of Gauss’s Law for magnetic field

  1. $\oint {} $ – Reminds that the integral is been taken over the closed surface
  2. $S$ – It reminds that it is a surface integral not a volume or line integral
  3. ${\overrightarrow B }$ The magnetic field – A magnetic field is the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude; as such it is a vector field (For more information visit this link ). Magnetic fields may be represented using field lines whose density in a plane perpendicular to the line direction is proportional to the strength of the field.
  4. $\oint\limits_S {\overrightarrow B  \cdot \widehat nda} $ Magnetic Flux – Magnetic flux through any surface is is the amount of magnetic field flowing through the surface. Calculating this quantity depends on the situations like
    • ${\varphi _B} = \left| {\overrightarrow B } \right| \times ({\text{surface area)}}$ when $ \overrightarrow B $ is uniform and perpendicular to the surface $S$
    • ${\varphi _B} = \overrightarrow B \cdot \widehat n \times ({\text{surface area)}}$ when $ \overrightarrow B $ is uniform and at an angle to the surface $S$
    • ${\varphi _B} = \int\limits_S {\overrightarrow B \cdot \widehat nda} $ when $ \overrightarrow B $ is nonuniform and at an variable angle to the surface $S$

    Magnetic flux is a scalar quantity and the units of magnetic flux is ‘‘webers’’ abbreviated as Wb. The magnetic flux through a surface may be considered to be the number of magnetic field lines penetrating that surface and you must established a relationship between the number of lines you draw and the strength of the field. When considering magnetic flux through a closed surface, it is important that you remember that surface penetration is taking both ways that is outward flux and inward flux have opposite signs.Thus equal amounts of outward (positive) flux and inward (negative) flux will cancel, producing zero net flux. The outward and inward magnetic flux must be equal and opposite through any closed surface. The number of field lines entering the volume enclosed by the surface is exactly equal to the number of field lines leaving that volume. This mean that the net magnetic flux through any closed surface must always be zero

So, the net magnetic flux passing through any closed surface must be zero because magnetic field lines always form complete loops.

Our next article would be about Differential form of Gauss’s Law for magnetic fields




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