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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?

enter image description here

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.

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  • $\begingroup$ The magnetic field is not constant in space. It has different values in different locations. $\endgroup$
    – Mark H
    Jan 31, 2020 at 9:59
  • $\begingroup$ Hint: think of the direction of rotation in each semicircle. $\endgroup$ Jan 31, 2020 at 11:03
  • $\begingroup$ Where does the y/|y| come from? $\endgroup$ Jan 31, 2020 at 11:07

2 Answers 2

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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.

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  • $\begingroup$ Thank you very much! $\endgroup$ Jan 31, 2020 at 18:38
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There are two different, opposite and with the same magnitude magnetic fields on upper and lower half-spaces.

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