A positively charged particle starts at origin with velocity v₀ making 60° with the positive y-axis. Fields E⃗ = −Eĵ and B⃗ = Bk̂ are uniform. For the particle to return to origin, B₍min₎ = nπE/v₀. Determine n. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A positively charged particle starts at origin with velocity v₀ making 60° with the positive y-axis. Fields E⃗ = −Eĵ and B⃗ = Bk̂ are uniform. For the particle to return to origin, B₍min₎ = nπE/v₀. Determine n. [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — For uniform B⃗ along z, write components of Lorentz force: m dvₓ/dt = qBvᵧ, and m dvᵧ/dt = −qBvₓ − qE. Step 2 — The drift velocity of the crossed E and B fields is along the x-axis with speed E/B. Superpose cyclotron motion of angular frequency ω = qB/m. Step 3 — Set x(t) = 0 and y(t) = 0 for the first nonzero return time and minimize B for the specified 60° launch direction. Step 4 — This yields B₍min₎ = 2πE/v₀, so n = 2.