Uniform electric and magnetic fields of magnitudes 1 V/m and 1 T are both along +y. A particle of mass 1 kg and charge 1 C starts at the origin with velocity 1 m/s along +x. At t=π s its coordinates are (X,Y,Z). Find (XY+YZ+ZX)·(5/π²).
Physics · Magnetic effect of current
JEE Advanced

Question

Uniform electric and magnetic fields of magnitudes 1 V/m and 1 T are both along +y. A particle of mass 1 kg and charge 1 C starts at the origin with velocity 1 m/s along +x. At t=π s its coordinates are (X,Y,Z). Find (XY+YZ+ZX)·(5/π²).

Physics · Magnetic effect of current

G1 Physics · Accepted Answer

The y-motion has constant acceleration qE/m=1, so y=π²/2. The x-z motion is cyclotron motion with ω=qB/m=1 and initial vₓ=1; at t=π the particle returns to x=0 with z=2. Hence XY+YZ+ZX=π², and multiplying by 5/π² gives 5.