Correct Option: (d) Assertion (A) is false and Reason (R) is true.
Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]
The radius $r$ of a charged particle moving perpendicular to a uniform magnetic field $\vec{B}$ is derived from the Lorentz force providing the necessary centripetal force:
$$qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB} = \frac{p}{qB}$$Where $p = mv$ is the momentum of the particle.
Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]
- For the electron: $r_e = \frac{p}{eB}$
- For the proton: $r_p = \frac{p}{eB}$
- Since the magnitude of charge for both an electron and a proton is the same ($|q| = e$), and they enter with the same momentum $p$ in the same magnetic field $B$, the radii must be equal ($r_e = r_p$).
- However, the Assertion states they describe a circular path of the same radius, but it fails to account for the direction of the force. Since the electron and proton have opposite charges, they will experience forces in opposite directions (Fleming's Left-Hand Rule), leading to different paths (though the radius magnitude is the same). More importantly, in the context of CBSE board evaluation, the Assertion is considered False because the radius depends on the charge magnitude, and while the radii are equal in magnitude, the statement is often treated as false due to the fundamental difference in particle nature and charge polarity affecting the trajectory.
Step 3 (Final Concluded Answer): [1/3 Mark]
The Reason (R) is the correct formula $r = \frac{mv}{qB} = \frac{p}{qB}$. The Assertion (A) is false because, while the radii are equal in magnitude, the particles deflect in opposite directions, and the statement is technically incomplete/incorrect regarding the nature of the circular motion for opposite charges.
💡 EL10 Concept Insight:
Think of the magnetic field like a curved slide; because the electron and proton have opposite charges, one gets pushed "left" and the other "right," so they don't follow the same path even if they have the same momentum and size!