A proton, a deuteron and an α particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is _________ and their speed is ________, in the ratio.

CONCEPT:

• The momentum is defined as the product of mass and velocity and it is written as;

            p = mv

           Here p is momentum, m is the mass and v is the velocity.

• The Lorentz force is written as;

           F = q (v × B)    or, F = qvB sinθ 

CALCULATIONS:

Let " m " be the mass of the proton and "q" is the charge of the proton.

As we know the deuteron is have twice the mass of the proton.

and the α  particle is having four times the mass of the proton and twice the charge of the proton.

As we know that momentum is written as;

p = mv ⇒v = 

For proton we have;

   -----(1)

For deuteron we have;

⇒     ----(2)

and for  particle we have;

⇒    ----(3)

Now, take the ratio of equations (1), (2), and (3) we have;

⇒  = 4 ∶ 2 ∶ 1

Now, the magnetic force is written as;

F = qvB

and we know v = 

∴ F = 

Now, for proton we have;

   ----(4)

For Deuteron we have;

⇒    ----(5)

and for α particle we have;

⇒ 

⇒     ----(6)

The ratio of equations (4), (5), and (6) we have;

⇒ 

⇒  = 2 ∶ 1 ∶ 1

Hence, option 1) is the correct answer.