A particle of charge q, mass m and kinetic energy E enters in magnetic field perpendicular to its velocity and undergoes a circular arc of radius (r). Which of the following curves represents the variation of r with E ?

Solution

Physics sir · 4 steps

Write the radius in terms of momentum, then momentum in terms of energy: r ∝ √E.

1. Radius of the arc. The magnetic force qvB supplies the centripetal force mv²/r. So r = mv/qB = p/qB, where p is the momentum.
2. Momentum from energy. Kinetic energy is E = p²/2m, so p = √2mE. Substitute this into the radius.

   r = √2mE/qB ⇒ r ∝ √E

3. Shape of the graph. For r ∝ √E, r = 0 when E = 0, and r keeps increasing as E increases. Its slope keeps falling, though: four times the energy gives only twice the radius. So the curve starts at the origin and bends downward, like a parabola lying on its side.
4. So. Graph (a) is the only one that rises from the origin and bends downward. It is the r versus E curve.

The idea to keep. In a magnetic field, r = p/qB and p = √2mE. Always convert to momentum first, then read the power of E to get the shape of the graph.

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