Rankers Physics

Force Acting on Moving Charges: Practice Problem & Solution

An alternating electric field, of frequency $\nu$, is applied across the dees (radius $= R$) of a cyclotron that is being used to accelerate protons ($\text{mass} = m$). The operating magnetic field ($B$) used in the cyclotron and the kinetic energy ($K$) of the proton beam, produced by it, are given by: (2012 Pre)
$B = \frac{mv}{e}$ and $K = 2m\pi^2\nu^2R^2$
$B = \frac{2\pi mv}{e}$ and $K = m^2\pi^2\nu^2R^2$
$B = \frac{2\pi m\nu}{e}$ and $K = 2m\pi^2\nu^2R^2$
$B = \frac{mv}{e}$ and $K = m^2\pi\nu R^2$

Solution Explained:

To solve this problem, we apply the core principles of Force Acting on Moving Charges. Understanding the underlying formula is key to arriving at the correct answer below:

Cyclotron frequency is $\nu = \frac{eB}{2\pi m}$, giving $B = \frac{2\pi m\nu}{e}$. Maximum kinetic energy is $K = \frac{e^2B^2R^2}{2m} = 2m\pi^2\nu^2R^2$.

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