onsider a proton moving in a three-dimensional space under the influence of both a uniform magnetic field and a non-uniform electric field. The magnetic field 𝐵 B is directed along the z-axis with a magnitude 𝐵 0 B 0 ​ . The electric field 𝐸 E varies with position and is given by 𝐸 ( 𝑥 , 𝑦 , 𝑧 )

𝐸 0 ( 𝑥 𝑎 2 𝑖 ^ + 𝑦 𝑏 2 𝑗 ^ + 𝑧 𝑐 2 𝑘 ^ ) E(x,y,z)=E 0 ​ ( a 2

x ​

i ^ + b 2

y ​

j ^ ​ + c 2

z ​

k ^ ), where 𝐸 0 E 0 ​ , 𝑎 a, 𝑏 b, and 𝑐 c are constants.

The proton starts from rest at the origin ( 0 , 0 , 0 ) (0,0,0) at 𝑡

0 t=0. Assuming that the proton's motion can be described by the non-relativistic equations of motion, find the complete trajectory 𝑟 ( 𝑡 ) r(t) of the proton as a function of time, taking into account the Lorentz force 𝐹

𝑞 ( 𝐸 + 𝑣 × 𝐵 ) F=q(E+v×B). Additionally, determine the time-dependent expressions for the velocity components 𝑣 𝑥 ( 𝑡 ) v x ​ (t), 𝑣 𝑦 ( 𝑡 ) v y ​ (t), and 𝑣 𝑧 ( 𝑡 ) v z ​ (t), and the total distance traveled by the proton from 𝑡

0 t=0 to 𝑡

𝑇 t=T, where 𝑇 T is the time at which the proton first returns to the 𝑥 𝑦 xy-plane.