The Field Keeps Me Down

Relevant wiki: Lorentz force law (mixed fields)

With the FBD we can find out that the magnitude of the normal is given by: $N=cos(\theta)\cdot mg + qB_{0}\dot{x}$ Then, the value of the friction force equals: $|f_{r}|=\mu (cos(\theta)\cdot mg + qB_{0}\dot{x})$ So the net force, using the Newton's Second Law of Motion , will give us the next differential equation: $m\ddot{x}=sin(\theta)\cdot mg - \mu (cos(\theta)\cdot mg + qB_{0}\dot{x})$ $\ddot{x}+\frac{\mu qB_{0}}{m}\dot{x}=g(sin(\theta)-\mu cos(\theta))$ Which gives us as solution for the velocity: $\dot{x}=\frac{mg}{\mu qB_{0}}(sin(\theta) - \mu cos(\theta))(1-e^{-\frac{\mu qB_{0}}{m}t})$ As the magnetic force, by Lorentz Law , is proportional to the velocity, the maximum magnetic force will be reached at the moment of maxmum velocity, which is $\dot{x}_{max}=\frac{mg}{\mu qB_{0}}(sin(\theta)-\mu cos(\theta))$ . So the maximum magnetic force equals: $F_{Bmax}=\frac{mg}{\mu}(sin(\theta)-\mu cos(\theta))$ Then: $sin^2(\theta)+cos^2(\theta)=1$