(LO3)Q16) Two charges with the masses m_(1)=4m_(2) and velocities V_(2)=4V_(1) enter to a uniform magnetic field B. If q_(2)=5q_(1) Find the ratio of the circulation radiures (r_(1)/r_(2)) A)4 B)! C)2 D)3 E)S

To find the ratio of the circulation radii $(r_{1}/r_{2})$, we can use the formula for the radius of the circular motion of a charged particle in a magnetic field:<br /><br />$r = \frac{mv}{qB}$<br /><br />Given that $m_{1}=4m_{2}$, $V_{2}=4V_{1}$, and $q_{2}=5q_{1}$, we can substitute these values into the formula:<br /><br />$r_{1} = \frac{m_{1}V_{1}}{q_{1}B}$<br />$r_{2} = \frac{m_{2}V_{2}}{q_{2}B}$<br /><br />Now, we can find the ratio of the circulation radii:<br /><br />$\frac{r_{1}}{r_{2}} = \frac{\frac{m_{1}V_{1}}{q_{1}B}}{\frac{m_{2}V_{2}}{q_{2}B}}$<br /><br />Substituting the given values:<br /><br />$\frac{r_{1}}{r_{2}} = \frac{\frac{4m_{2}V_{1}}{q_{1}B}}{\frac{m_{2}V_{2}}{5q_{1}B}}$<br /><br />Simplifying the expression:<br /><br />$\frac{r_{1}}{r_{2}} = \frac{4V_{1}}{V_{2}} \cdot \frac{5q_{1}}{q_{2}}$<br /><br />Since $V_{2}=4V_{1}$ and $q_{2}=5q_{1}$, we can substitute these values:<br /><br />$\frac{r_{1}}{r_{2}} = \frac{4V_{1}}{4V_{1}} \cdot \frac{5q_{1}}{5q_{1}}$<br /><br />Simplifying further:<br /><br />$\frac{r_{1}}{r_{2}} = 1$<br /><br />Therefore, the ratio of the circulation radii $(r_{1}/r_{2})$ is 1.<br /><br />The correct answer is E) 1.