A particle of mass m and charge q has an initial velocity v⃗ = v₀ĵ . If an electric field E⃗ = E₀î and magnetic field B⃗ = B₀î act on the particle, its speed will double after a time:

Solution

Physics sir · 4 steps

The magnetic field only turns the sideways velocity, so only the electric field can add speed, along x.

1. Split the velocity. Both fields point along x. Track the x part of the velocity separately from the part in the y-z plane.
2. Magnetic field. The force qv⃗ × B₀î is always perpendicular to v⃗, so it does no work. It makes the y-z part rotate in a circle, and that part keeps its size v₀ at all times.
3. Electric field. The force qE₀ along x gives a constant acceleration qE₀/m. Starting from zero, vₓ = qE₀/mt.

   v² = vₓ² + v₀² = (2v₀)² ⇒ vₓ = √3 v₀

4. So. qE₀/mt = √3 v₀, which gives t = (√3 mv₀)/qE₀. That is option (c).

The idea to keep. A magnetic force never changes speed. Split the velocity along and across B⃗: the part across stays the same size, and any field along B⃗ changes only the part along it.

Answer from the NTA final answer key. Solution by PrepWiser.