A proton and an alpha particle moving with same kinetic energy enter in the region of uniform magnetic field perpendicular to it. The ratio of radii of their trajectories will be:

Concept:

The radius of the trajectory of a charged particle moving in a uniform magnetic field is given by the formula:

r = (mv)/(qB)

where:

r is the radius of the trajectory
m is the mass of the particle
v is the velocity of the particle
q is the charge of the particle
B is the magnetic field strength
The ratio of the radii of the trajectories of a proton and an alpha particle can be found by dividing their respective radii:

r_(proton) / r_(alpha) = [(m_protonv_proton)/(q_protonB)] / [(m_alphav_alpha)/(q_alphaB)] ------(1)

Since both particles have the same kinetic energy, we can use the fact that the kinetic energy of a particle is given by:

K = (1/2)mv²

Thus, we have:

v_proton = \(\sqrt\frac{2K}{m_(proton)}\)
v_alpha =\(\sqrt\frac{2K}{m_(alpha)}\)

Substituting these expressions into the ratio of radii equation(1),and Plugging in the values for the masses and charges of the particles, we get

r(proton) / r(alpha) = [(1/4)*(2/1)]*sqrt(4/1) = 1/2 * 2 = 1

Therefore, the ratio of the radii of the trajectories of the proton and the alpha particle is 1:1 or simply 1.

This means that the radii of their trajectories will be the same.

The correct answer is option (1)