Divergence of B, circular current loop

In summary, the conversation discusses the concept of magnetostatics and its relation to steady current. It is proven that the equations ∇ * B = 0 and ∇ X B = Mu * J hold for magnetostatics, including closed wire loops with constant current. The magnetic field on the z-axis above the loop around the origin is given by B = (Mu * I * R^2)/(2 * (R^2 + z^2)^(3/2)) in the z-hat direction. The participants also mention the use of partial derivatives and cylindrical coordinates to verify the divergence of B is not zero.
  • #1
pobro44
11
0

Homework Statement


[/B]
∇ * B = 0 and ∇ X B = Mu * J. This is proved to hold not only for infinite wires but for magnetostatics in general.

Magnetostatics = steady current

Closed wire loop with constant current is certainly a magnetostatics example.

Magnetic field on z axis above loop around origin is: B = (Mu* I * R^2)/(2 * (R^2 + z^2)^(3/2)) in z hat direction

Homework Equations



Partial derivative with respect to z gives a non zero answer. Divergence is not zero. I am missing something obvious but fail to see what.

The Attempt at a Solution

 
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  • #2
Divergence is not just the partial derivative along z. Really think about the meaning of ##\frac{\partial \mathbf{B}_x}{\partial x}## and ##\frac{\partial \mathbf{B}_y}{\partial y}##.
 
  • #3
Can also use cylindrical coordinates to verify ∇⋅B = 0. Look up the formula for div in cylindrical coordinates and apply to the problem.
 

1. What is divergence of B in a circular current loop?

The divergence of B in a circular current loop refers to the change in the magnetic field strength at a given point due to the flow of current around the loop. It is a measure of how much the magnetic field lines are spreading out or converging at that point.

2. How is the divergence of B calculated in a circular current loop?

The divergence of B in a circular current loop can be calculated using the equation div B = μ0I/2πr, where μ0 is the permeability of free space, I is the current flowing through the loop, and r is the distance from the center of the loop to the point of interest.

3. What is the significance of divergence of B in a circular current loop?

The divergence of B in a circular current loop is significant because it determines the strength and direction of the magnetic field at a given point. It also plays a crucial role in understanding electromagnetic induction and the behavior of magnetic materials.

4. How does the divergence of B change with distance from a circular current loop?

The divergence of B decreases as the distance from a circular current loop increases. This is because the magnetic field lines spread out as they move away from the loop, resulting in a decrease in the magnetic field strength at farther distances.

5. Can the divergence of B be negative in a circular current loop?

Yes, the divergence of B can be negative in a circular current loop. This indicates that the magnetic field lines are converging at that point, rather than spreading out. This can occur when the direction of the current flow and the orientation of the loop are such that the magnetic field lines are directed towards the center of the loop.

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