An electron enters with a velocity $\vec{v} = v_0 \hat{i}$ into a cubical region (faces parallel to coordinate planes) in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in a plane parallel to the $xy$-plane. Suggest a configuration of fields $\vec{E}$ and $\vec{B}$ that can lead to this.

(A) The electron enters the region with velocity $\vec{v} = v_0 \hat{i}$.
To have a spiral motion in a plane parallel to the $xy$-plane,the magnetic force must provide the centripetal force for circular motion in the $xy$-plane. This requires a magnetic field $\vec{B}$ along the $z$-axis,i.e.,$\vec{B} = B_0 \hat{k}$.
The magnetic force is $\vec{F}_m = -e(\vec{v} \times \vec{B}) = -e(v_0 \hat{i} \times B_0 \hat{k}) = e v_0 B_0 \hat{j}$.
For the electron to spiral,its speed must change,which is caused by an electric field $\vec{E}$. Since the electron is moving in the $xy$-plane and the spiral is expanding or contracting,an electric field component in the $xy$-plane is required. Specifically,an electric field $\vec{E} = E_0 \hat{i}$ would accelerate the electron,increasing its speed $v$,which in turn increases the radius of the circular path $r = \frac{mv}{eB}$,resulting in a spiral trajectory.