Question

Two ions of masses 4 amu and 16 amu have charges +2e and +3e respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then :

• A. lighter ion will be deflected more than heavier ion
• B. lighter ion will be deflected less than heavier ion
• C. both ions will be deflected equally
• D. no ion will be deflected

Hint:

Deflection is inversely proportional to the radius of the circular path (\(r\)). Always express the radius in terms of the given quantities. In this case, since kinetic energy (\(K\)) is given, use the formula \(r = \frac{\sqrt{2mK}}{qB}\). Then analyze the proportionality \(r \propto \frac{\sqrt{m}}{q}\).

Answer 1

The Correct Option is A

To solve this problem, we need to understand how charged particles behave in a magnetic field. The path of a charged ion in a magnetic field is circular, and the radius of this path is determined by the magnetic force acting on the ion. The formula for the radius of the path is given by:

\(r = \frac{mv}{qB}\) 

where \(r\) is the radius of the circular path, \(m\) is the mass of the ion, \(v\) is the velocity of the ion, \(q\) is the charge of the ion, and \(B\) is the magnetic field strength.

Since the kinetic energy (\(KE\)) of both ions is the same, we know that:

\(\text{KE} = \frac{1}{2}mv^2\)

This implies:

\(v = \sqrt{\frac{2 \times \text{KE}}{m}}\)

Substituting the velocity (\(v\)) in the formula for radius (\(r\)):

\(r = \frac{m\sqrt{\frac{2 \times \text{KE}}{m}}}{qB} = \frac{\sqrt{2m \times \text{KE}}}{qB}\)

Now, compute the radius for both ions:

1. For the lighter ion (mass = 4 amu, charge = +2e):
2. For the heavier ion (mass = 16 amu, charge = +3e):

Now we compare the radii to determine which ion is deflected more. In doing so, we look at the expression \(\frac{\sqrt{2m \times \text{KE}}}{q}\). The ion with the smaller radius is deflected more:

• The ratio of masses forms a key point: the lighter ion (mass = 4 amu) has a higher charge-to-mass ratio (\(\frac{2}{4}\)) compared to the heavier ion (\(\frac{3}{16}\)).
• This means that the lighter ion has a greater curvature in its path compared to heavier ion under the same kinetic energy and magnetic field conditions.

Thus, the correct choice is that the lighter ion will be deflected more than the heavier ion. Therefore, the statement "lighter ion will be deflected more than heavier ion" is correct.