Question

Two particles \(\mathrm{X}\) and \(\mathrm{Y}\) having equal charges, after being accelerated through the same potential difference, enter a region of uniform mag. field and describe circular path of radius \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) respectively. The ratio of mass of \(\mathrm{X}\) to that of \(\mathrm{Y}\) is (a) \(\sqrt{\left(r_{1} / \mathrm{r}_{2}\right)}\) (b) \(\left(\mathrm{r}_{2} / \mathrm{r}_{1}\right)\) (c) \(\left(\mathrm{r}_{1} / \mathrm{r}_{2}\right)^{2}\) (b) \(\left(\mathrm{r}_{1} / \mathrm{r}_{2}\right)\)

Short answer

The ratio of the mass of particle X to that of particle Y is given by: \[\frac{m_x}{m_y} = \frac{r_2}{r_1}\] which corresponds to the option (b).

Step-by-step solution

Determining the velocity of the particles using the potential difference

The kinetic energy gained by the particles after being accelerated through the potential difference V is given by: \[K.E. = qV\] where q is the charge of the particles. The kinetic energy of a particle is also given by: \[K.E. = \frac{1}{2}m\upsilon^2\] where m is the mass of the particle and 𝜈 is its velocity. Equating both expressions for kinetic energy, we can determine the velocity of the particles as follows: \[qV = \frac{1}{2}m\upsilon^2\]

Apply the Lorentz force formula to the particles

The Lorentz force acting on a charged particle in a magnetic field is given by: \[F = q\upsilon B\] where B is the magnetic field. Since the charged particles are moving in a circular path, the centripetal force required is given by: \[F_c = \frac{m\upsilon^2}{r}\] Equating the Lorentz force with the centripetal force, we get: \[q\upsilon B = \frac{m\upsilon^2}{r}\]

Finding the ratio of the masses

We need to find the ratio of the masses of the particles X and Y. Let's denote these masses as 𝑚𝑥 and 𝑚𝑦, respectively. We have the following equations for the particles X and Y from step 2: \[q\upsilon_x B = \frac{m_x\upsilon_x^2}{r_1}\] \[q\upsilon_y B = \frac{m_y\upsilon_y^2}{r_2}\] Now, 𝑞 and 𝐵 are equal for both particles, so we can divide the first equation by the second equation to get the ratio of the masses: \[\frac{m_x\upsilon_x^2/r_1}{m_y\upsilon_y^2/r_2} = 1\] Rearranging the equation to find the ratio (m_x/m_y), we get: \[\frac{m_x}{m_y} = \frac{\upsilon_x^2/r_1}{\upsilon_y^2/r_2}\] Now, using the formula for the velocity we derived in step 1, substitute the velocities in terms of potential difference: \[\upsilon_x^2 = \frac{2qV}{m_x}\] \[\upsilon_y^2 = \frac{2qV}{m_y}\] Substitute these values into the ratio equation: \[\frac{m_x}{m_y} = \frac{\left(\frac{2qV}{m_x}\right)/r_1}{\left(\frac{2qV}{m_y}\right)/r_2}\] Simplify the equation: \[\frac{m_x}{m_y} = \frac{2qVr_2}{2qVr_1}\] The term \(2qV\) cancels out from both numerator and denominator, leaving us with the final answer: \[\frac{m_x}{m_y} = \frac{r_2}{r_1}\] So, the ratio of the mass of particle X to that of particle Y is given by: \[\frac{m_x}{m_y} = \frac{r_2}{r_1}\] which corresponds to the option (b).