Question

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 ?

• A. A straight line
• B. A circle
• C. A parabola
• D. An ellipse

Answer 1

Answer: (C)

A parabola

Answer 2

Here's the breakdown of why the answer is a parabola:

1. Magnetic Force and Circular Motion: When a charged particle (charge q, mass m, velocity v) enters a magnetic field B perpendicularly, it experiences a magnetic force (F = qvB) that acts as the centripetal force, causing circular motion.

2. Equating Forces:

   • Magnetic force: $F_B = qvB$
   • Centripetal force: $F_c = \frac{mv^2}{r}$
   • Equating: $qvB = \frac{mv^2}{r}$

3. Radius (r) Formula: Solving for the radius r: $r = \frac{mv}{qB}$

4. Kinetic Energy (E) Relation:

   • Kinetic energy: $E = \frac{1}{2}mv^2$
   • Solving for v: $v = \sqrt{\frac{2E}{m}}$

5. Substitute v into r Formula: $r = \frac{m}{qB} \sqrt{\frac{2E}{m}} = \frac{\sqrt{2mE}}{qB}$

6. Relationship between r and E: The equation $r = \frac{\sqrt{2mE}}{qB}$ shows that r is proportional to the square root of E: $r \propto \sqrt{E}$

   This implies that if you plot r on the y-axis and E on the x-axis, the graph will be a parabola (specifically, the upper half of a parabola opening along the positive x-axis).