Exploring the Lorentz Force: What Goes Wrong?

[Gabriele Pinna]

We know that Lorentz Force is a centripetal force so:
$$qVBsenx=m\frac{v^2}{r}$$ and the value of V is constant (but not the direction)
But if we write:
$$qvBsenx=m\frac{dV}{dt}$$
we obtain that v is exponential.
What is wrong ?

 

[vanhees71]

I guess you want to solve the nonrelativistic equation of motion for a particle in a constant magnetic field. Let's put $\vec{B}=B \vec{e}_z$. Then the equations of motion read (in SI units)
$$m \frac{\mathrm{d} \vec{v}}{\mathrm{d}t} = q \vec{v} \times \vec{B}.$$
In components you have
$$\begin{pmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \end{pmatrix}=\frac{qB}{m} \begin{pmatrix} v_y \\ -v_x \\ 0 \end{pmatrix}=\omega \begin{pmatrix} v_y \\ -v_x \\ 0\end{pmatrix}.$$
Now you can integrate the equations of motion quite easily. I give some hints:

The equation for $v_z$ is very simple. This you should be able to solve very easily.

To solve also for $v_x$ and $v_y$, just take the time derivative of the equation for $v_x$ and use the 2nd equation to eliminate $v_y$. Then you have a equation for $v_x$ alone, which is quite easy to solve too.

Once you have $\vec{v}$ just take one more integration to get $\vec{x}$.

 

[Orodruin]

You are trying to write a vector equation as a single scalar equation.