A particle with charge $q$ and mass $m$ is moving in a region containing a uniform electric field $E$ and a uniform magnetic field $B$, both directed along the $z$-axis. If the particle starts from the origin with velocity $v$ in the $xy$-plane, what is the path of the particle?

Solution

1. The magnetic field $B$ along the $z$-axis exerts a Lorentz force that causes the particle to move in a circle within the $xy$-plane with a constant angular frequency $\omega =qB/m$.
2. The electric field $E$ along the $z$-axis exerts a constant force $F_z=qE$, resulting in a constant acceleration $a_z=qE/m$ in the $z$-direction.
3. The velocity in the $z$-direction is given by $v_z(t)=(qE/m)t$, which increases linearly with time.
4. The superposition of uniform circular motion in the $xy$-plane and linearly increasing velocity along the $z$-axis describes a helix with a pitch that increases over time.
5. Hence the answer is (A).