A deuteron and a proton moving with equal kinetic energy enter into a uniform magnetic field at right angle to the field. If r d and rₚ are the radii of their circular paths respectively, then the ratio r_d/rₚ will be √x : 1 where x is __________.

Solution

Physics sir · 4 steps

Equal kinetic energy, not equal speed: write the radius in terms of momentum, p = √2mK.

1. Radius from force. The magnetic force qvB provides the centripetal force mv²/r, so r = mv/qB = p/qB.
2. Bring in K. Kinetic energy is K = p²/2m, so p = √2mK. This gives the radius directly in terms of K.

   r = √2mK/qB

3. Compare the two. A deuteron has one proton and one neutron, so m_d = 2mₚ and q_d = qₚ = e. With the same K and B, r_d/rₚ = √(m_d/mₚ) = √2.
4. So. r_d : rₚ = √2 : 1, so x = 2.

The common slip: 4. Taking the speeds as equal makes r ∝ m, which gives r_d/rₚ = 2 = √4 and so x = 4. Here the kinetic energies are equal, so the deuteron moves slower.

The idea to keep. In a magnetic field r = √2mK/qB. For equal kinetic energy and equal charge, the radius goes as √m, not as m.

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