Question

A particle of charge $ q $, mass $ m $, and kinetic energy $ E $ enters in a 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.
• B.
• C.
• D.

Hint:

For a charged particle moving in a magnetic field, the radius of the circular path is proportional to the square root of its kinetic energy.

Answer 1

The Correct Option is D

Given: A particle with charge \( q \), mass \( m \), and kinetic energy \( E \) enters a magnetic field perpendicular to its velocity, moving in a circular arc of radius \( r \). The radius \( r \) is defined by \( r = \frac{mv}{qB} \). The kinetic energy is \( E = \frac{1}{2} mv^2 \). From this, the velocity \( v \) can be expressed as \( v = \sqrt{\frac{2E}{m}} \). Substituting \( v \) into the radius formula yields \( r = \frac{m\sqrt{\frac{2E}{m}}}{qB} = \frac{\sqrt{2mE}}{qB} \). Consequently, \( r \) is directly proportional to the square root of \( E \), indicated by \( r \propto \sqrt{E} \). This establishes a curved relationship between \( r \) and \( E \), as depicted in option (4).
Final Answer (4)