Calculating the curvature drift in a plasma

[TimeRip496]

How do i get this equation,
$$\frac{db}{dl}=(\hat{B}.∇)\hat{b}$$

This equation is a vector whose direction is towards the centre of the circle which most closely approximates the magnetic field-line at a given point, and whose magnitude is the inverse of the radius of this circle.

However I have no idea how to obtain the above and the source basically state it as by definition. Where is this equation obtained from?

Source: https://ocw.mit.edu/courses/nuclear...a-physics-i-fall-2003/lecture-notes/chap2.pdf page 8

 

[AndyW]

Thank you for your question. The equation you have mentioned is known as the magnetic field line equation. This equation is derived from the fundamental laws of electromagnetism, namely Maxwell's equations. Specifically, it is derived from one of Maxwell's equations, known as the curl equation, which relates the curl of a vector field to its sources.

In order to obtain this equation, we first start with the definition of the magnetic field, which is given by the vector potential, A. The magnetic field can be written as the curl of the vector potential, B=∇×A. Using this definition and Maxwell's equations, we can derive the magnetic field line equation.

I would recommend studying the derivation of this equation in a textbook or through online resources, as it involves some mathematical manipulation and may be difficult to explain fully in a forum post. However, I can provide some key points to help you understand the derivation:

1. Start with the definition of the magnetic field, B=∇×A.
2. Use the curl equation of Maxwell's equations to express the curl of B in terms of the curl of A.
3. Use the vector triple product identity to simplify the expression.
4. Apply the definition of the magnetic field line direction, which is perpendicular to the magnetic field.
5. Simplify the expression further to obtain the final equation.

I hope this helps you understand the derivation of the magnetic field line equation. If you have any further questions, please feel free to ask. Good luck with your studies!