An electron and a proton, both having the same kinetic energy, enter a region of uniform magnetic field, in a plane perpendicular to the field. If their masses are denoted by `m_(e)` and `m_(p)` respectively, then the ratio of the radii (electron to proton) of their circular orbits is

Options

A. `sqrt((m_(p))/(m_(e)))`
B. `sqrt((m_(e))/(m_(p)))`
C. `(m_(e))/(m_(p))`
D. 1

Explanation

To find the ratio of the radii of the circular orbits of an electron and a proton moving in a uniform magnetic field with the same kinetic energy, we can follow these steps:

### Step 1: Understand the relationship between radius, mass, charge, and velocity in a magnetic field
When a charged particle moves in a magnetic field, the radius \( r \) of its circular path is given by the formula:
\[ r = \frac{mv}{Bq} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field strength,
- \( q \) is the charge of the particle.

### Step 2: Express velocity in terms of kinetic energy
The kinetic energy \( KE \) of a particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
From this, we can express \( v \) as:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 3: Substitute velocity into the radius formula
Substituting the expression for \( v \) into the radius formula gives:
\[ r = \frac{m \cdot \sqrt{\frac{2 \cdot KE}{m}}}{Bq} \]
This simplifies to:
\[ r = \frac{\sqrt{2 \cdot KE \cdot m}}{Bq} \]

### Step 4: Write the radius for both the electron and proton
Let’s denote:
- \( m_e \) and \( q_e \) for the electron (mass and charge),
- \( m_p \) and \( q_p \) for the proton (mass and charge).

Since both particles have the same kinetic energy, we can write:
- Radius of the electron \( r_e \):
\[ r_e = \frac{\sqrt{2 \cdot KE \cdot m_e}}{B \cdot e} \]
- Radius of the proton \( r_p \):
\[ r_p = \frac{\sqrt{2 \cdot KE \cdot m_p}}{B \cdot e} \]

### Step 5: Find the ratio of the radii
Now, we can find the ratio of the radii:
\[ \frac{r_e}{r_p} = \frac{\sqrt{2 \cdot KE \cdot m_e}}{B \cdot e} \cdot \frac{B \cdot e}{\sqrt{2 \cdot KE \cdot m_p}} = \frac{\sqrt{m_e}}{\sqrt{m_p}} = \sqrt{\frac{m_e}{m_p}} \]

### Conclusion
Thus, the ratio of the radii of the circular orbits of the electron to the proton is:
\[ \frac{r_e}{r_p} = \sqrt{\frac{m_e}{m_p}} \]