The ratio of de - Broglie wavelength of `alpha `- particle to that of a proton being subjected to the same magnetic field so that the radii of their path are equal to each other assuming the field induction vector `vec(B)` is perpendicular to the velocity vectors of the `alpha` - particle and the proton is

Explanation

To solve the problem of finding the ratio of the de Broglie wavelength of an alpha particle to that of a proton when both are subjected to the same magnetic field and have equal radii of their paths, we can follow these steps:

### Step 1: Understand the relationship between radius, mass, charge, and velocity in a magnetic field.
The radius \( r \) of the circular path of a charged particle in a magnetic field is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
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