Rate of change of angle between a particle and a field

[James R]

I am wondering if anybody here can help me with a proof.

A particle is moving with velocity $\vec{v}$ in a field $\vec{F}$ (which could be an electric or magnetic field, for example). The velocity can change with time as the particle moves, but the field is static in space. The particle sees the field change as it moves from one position to another in space.

Now, I am interested in the time rate of change of the angle between the velocity vector and the field direction as the particle moves.

In other words, if at some point along the particle's trajectory the angle between F and v is $\theta$, then I need a vector expression for $d\theta / dt$.

Here's an expression I think is correct:

$$\frac{d\theta}{dt} = \frac{|\vec{F} \times (\vec{v} \cdot \nabla)\vec{F}|}{|\vec{F}|^2}$$

What I need is a derivation of this expression, or any kind of proof that it is in fact correct.

Can anybody help?

 

[James R]

LaTeX is not working at the moment, so here's a version without LaTeX formatting:

I am wondering if anybody here can help me with a proof.

A particle is moving with vector velocity v in a vector field F (which could be an electric or magnetic field, for example). The velocity can change with time as the particle moves, but the field is static in space. The particle sees the field change as it moves from one position to another in space.

Now, I am interested in the time rate of change of the angle between the velocity vector and the field direction as the particle moves.

In other words, if at some point along the particle's trajectory the angle between F and v is theta, then I need a vector expression for d(theta)/dt.

Here's an expression I think is correct:

d(theta)/dt = |F (cross product) (v.(gradient operator))F| / |F|^2

What I need is a derivation of this expression, or any kind of proof that it is in fact correct.

Can anybody help?