Question

A proton, deutron and an $\alpha$ - particle are accelerated by same potential, enter in uniform magnetic filed perpendicularly. Ratio of radii of circular path respectively :-

• A. $1:\sqrt{2}:\sqrt{2}$
• B. $2:2:1$
• C. $1:2:1$
• D. $1:1:1$

Answer 1

Answer: (A)

$1:\sqrt{2}:\sqrt{2}$

$\text{Given}$: Particles accelerated by same potential

$\text{To find}$: Ratio of radii

$\text{Solution}$ :

Mass and charge of proton is m and +e respectively

mass and charge of deuteron is 2m and +e respectively

mass and charge of alpha particle is 4m and +2e respect

Force due to magnetic field = $B\times q\times v$

This force is balanced by centripetal force in the field = $\frac{m{v}^2}{r}$

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

That is r = $\frac{m\times v}{B\times q}$

Therefore r ∝ $\frac{m\times v}{q}$ (as B is same) ................ (1)

All three particles are accelerated through the same field

$V\times q$ = $\frac{1}{2}m{v}^2$(Energy balance the energy under same potential and is converted to kinetic)

As V is same therefore v ∝ $\sqrt{\frac{q}{m}}$

from eq (1)

r ∝ $\frac{(m\times \sqrt{\frac{q}{m}})}{q}$ ⇒ r ∝ $\sqrt{\frac{m}{q}}$

Therefore $r_1:r_2:r_3$ = 1:$\sqrt{2}:\sqrt{2}$

$\text{Hence A is the correct option.}$