# Lorentz Force and Circular motion - What is the magnetic field causing the movement? [closed]

I am given this problem:

A particle with mass $$m$$ and positive charge $$q$$ is moving in the following path on the $$x$$-$$y$$ plane. It's path consists of semicircles as shown below. The particle's velocity at the origin is $$V_0$$ in the $$\hat y$$ direction. What is the magnetic field causing this movement?

My thought was using the Lorentz force : $$F=q(V\times B)$$, from this I know that the magnetic field is in the $$\hat z$$ direction, and using circular motion: $$F=ma$$, $$a=\frac{V^2}R$$.

But this is not the classic circular motion.

• The magnetic field is not constant in space. It has different values in different locations. Jan 31 '20 at 9:59
• Hint: think of the direction of rotation in each semicircle. Jan 31 '20 at 11:03
• Where does the y/|y| come from? Jan 31 '20 at 11:07

The magnetic field changes its direction when it crosses the x-axis. As you said, one can use the formula $$qvB=\frac{ mv²}{r}$$ For one side (say +y side) the magnetic field must be along the +z-axis. When the particle crosses the x-axis after completing a semicircle the magnetic field has same magnitude but reverses sign. Hence one can write $$B= \left\{ \begin{array}{ll} - \frac{mv}{qr} & \quad y < 0 \\\frac{mv}{qr} & \quad y > 0 \end{array} \right.$$ Where the magnetic field is along the z-axis.

Hope this helps.

• Thank you very much! Jan 31 '20 at 18:38

There are two different, opposite and with the same magnitude magnetic fields on upper and lower half-spaces.