A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charge particle increases to 4 times its initial value. What will be the ratio of new radius to the original radius of circular path of the charge particle?

A charged particle in a cyclotron has its kinetic energy increased to 4 times its original value, and we seek the ratio of the new radius to the original radius. In a magnetic field, the radius of the circular path of a charged particle is given by:

$$r = \frac{mv}{qB}$$

Since $$KE = \frac{1}{2}mv^2$$, it follows that $$v = \sqrt{\frac{2KE}{m}}$$. Substituting this result into the expression for the radius, one obtains:

$$r = \frac{m}{qB}\sqrt{\frac{2KE}{m}} = \frac{\sqrt{2mKE}}{qB}$$, which implies that $$r \propto \sqrt{KE}$$.

With the kinetic energy increased by a factor of 4, the ratio of the new radius to the original radius becomes: $$\frac{r_{\text{new}}}{r_{\text{original}}} = \sqrt{\frac{KE_{\text{new}}}{KE_{\text{original}}}} = \sqrt{\frac{4KE}{KE}} = \sqrt{4} = 2$$, so that the new radius is twice the original radius, or $$2 : 1$$.

Thus, the correct answer is Option C: 2 : 1.