A2 Daily A Level Physics question
A proton and a deuteron are accelerated from rest through the same potential difference. They then both enter a region containing a uniform magnetic field, with their velocities perpendicular to the field direction. A deuteron has the same charge as a proton but approximately twice the mass. How does the radius of the deuteron's circular path, r_d, compare to the radius of the proton's path, r_p?
Answer
The correct answer is A.
Correct: A — r_d is approximately 1.4 times r_p.. The work done on each particle is qV, so they both gain the same kinetic energy (KE). The radius of a charged particle's path in a magnetic field is given by rearranging Bqv = mv²/r to get r = mv/Bq. Since momentum p = mv = √(2mKE), the radius can be expressed as r = √(2mKE)/Bq. As B, q, and KE are the same for both particles, the radius r is proportional to the square root of the mass, √m. Since the deuteron's mass is twice the proton's mass, its radius will be √2 ≈ 1.414 times larger. B is incorrect because although the kinetic energy and charge are the same, the radius depends on momentum (mv), not directly on kinetic energy; the difference in mass leads to different momenta. C is incorrect as this represents the inverse ratio (1/√2), which would imply the radius is inversely proportional to the square root of the mass. D is incorrect because it assumes the radius is directly proportional to the mass (r ∝ m), ignoring that the more massive deuteron is moving slower for the same kinetic energy (v ∝ 1/√m), which partially counteracts the effect of the larger mass.