An electron, a proton, a deuteron and an alpha particle, each having the same speed are in a region of constant magnetic field perpendicular to the direction of the velocities of the particles. The radius of the circular orbits of these particles are respectively Re, Rp, Rd and Rα . If follows that

Question

An electron, a proton, a deuteron and an alpha particle, each having the same speed are in a region of constant magnetic field perpendicular to the direction of the velocities of the particles. The radius of the circular orbits of these particles are respectively Re, Rp, Rd and Rα . If follows that (A) Re=Rp (B) Rp=Rd (C) Rd=Rα  (D) Rp=Rα

Tardigrade Admin · Accepted Answer

In perpendicular magnetic field magnetic force provided centripetal force.
By the relation

q v B = \frac{m v ^{2}}{r}

r = \frac{m v}{qB} r \propto \frac{m}{q}

Since,

\frac{m}{q}

ratio for deuteron and

α

-particle is the same.

(\frac{m}{q} = \frac{4}{2} = 2 f or α - p a r t i c l e)

an d (\frac{m}{q} = \frac{2}{1} = 2

for deuteron).
Hence, the radius is the same for both.

$R_{e} = R_{p}$

$R_{p} = R_{d}$

$R_{d} = R_{α}$

$R_{p} = R_{α}$

Re​=Rp​

Rp​=Rd​

Rd​=Rα​

Rp​=Rα​

In perpendicular magnetic field magnetic force provided centripetal force. By the relation qvB=rmv2​ r=qBmv​r∝qm​ Since, qm​ ratio for deuteron and α -particle is the same. (qm​=24​=2forα−particle) and(qm​=12​=2 for deuteron). Hence, the radius is the same for both.

$R_{e} = R_{p}$

$R_{p} = R_{d}$

$R_{d} = R_{α}$

$R_{p} = R_{α}$