Divergence and Curl of magnetic field
Divergence of Magnetic Field
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/divergence.jpeg?resize=930%2C723&ssl=1)
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](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/1-1.jpg?resize=524%2C104&ssl=1)
Therefore, the magnetic field at P due to the whole current loop is given by
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/2-1.jpg?resize=353%2C71&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/3-1.jpg?resize=371%2C98&ssl=1)
Taking divergence both sides, we get
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/4-1.jpg?resize=426%2C105&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/5-2.jpg?resize=526%2C60&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/6-1.jpg?resize=656%2C107&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/7.jpg?resize=811%2C64&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/8-1.jpg?resize=284%2C92&ssl=1)
We know curl of gradient is zero.
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/9-1.jpg?resize=461%2C91&ssl=1)
Using eqns. (4) and (5), eqn. (3) becomes
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/10-1.jpg?resize=204%2C73&ssl=1)
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](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/curl.jpeg?resize=930%2C572&ssl=1)
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](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/11-1.jpg?resize=300%2C112&ssl=1)
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](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/12-1.jpg?resize=472%2C67&ssl=1)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/13-1.jpg?resize=514%2C95&ssl=1)
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](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/14-1.jpg?resize=436%2C88&ssl=1)
From (2) and (3)
![Divergence and curl of magnetic field](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/07/15-1.jpg?resize=357%2C201&ssl=1)
Which is the differential form of ampere’s law or curl of a magnetic field.