Prove Divergence and Curl of magnetic field

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Divergence and Curl of magnetic field

Divergence of Magnetic Field

Divergence and curl of magnetic field

We know, the magnetic field produced by a current element Id L vector at a point P (x,y,z) whose distance from the current element r is given by

Divergence and curl of magnetic field

Therefore, the magnetic field at P due to the whole current loop is given by

Divergence and curl of magnetic field
Divergence and curl of magnetic field

Taking divergence both sides, we get

Divergence and curl of magnetic field
Divergence and curl of magnetic field
Divergence and curl of magnetic field
Divergence and curl of magnetic field
Divergence and curl of magnetic field

We know curl of gradient is zero.

Divergence and curl of magnetic field

Using eqns. (4) and (5), eqn. (3) becomes

Divergence and curl of magnetic field

Thus, divergence of B vector is zero.

Any vector whose divergence is zero is known as a solenoidal vector. Thus, magnetic field vector B vector is a solenoidal vector.

This is the proof of Divergence of magnetic field.

Curl of Magnetic Field

Divergence and curl of magnetic field

Let us consider a region of space in which currents are flowing, the current density J vector varies from point to point but is time-independent.

The total steady current I is given by

Divergence and curl of magnetic field

where J vector is the current density of an element dS vector of the surface S bounded by the closed path.

Divergence and curl of magnetic field
Divergence and curl of magnetic field

But according to Stokes theorem, the closed line integral of the B vector is equal to the surface integral of its curl.

Divergence and curl of magnetic field

From (2) and (3)

Divergence and curl of magnetic field

Which is the differential form of ampere’s law or curl of a magnetic field.

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