# The Ultimate problem!

**Electricity and Magnetism**Level 5

A particle with specific charge \(\text{q/m}\) moves in the region of space where there are uniform mutually perpendicular electric and magnetic fields with strength **E** (along \(y\) axis) and magnetic field **B** (along \(z\)-axis). It had zero initial velocity and was located at the origin. For non-relativistic case, find the law of motion \(x(t)\) and \(y(t)\) of the particle and hence, the shape of the trajectory.

Define

- \(P\) as the ratio of the area of the figure made with the \(x\)-axis when it again touches the \(x\)-axis,

- \(Q\) as the length of the "arch" of that section of the figure,

- \(R\) as the angular speed at which it reaches.

If the value of \( \frac{P}{Q\times R} \) equals to

\[\displaystyle \frac{a\pi {m}^{b}{E}^{c}{q}^{d}{B}^{e}}{f}\]

give your answer as \(\displaystyle a + b + c + d + e + f\).