Three charged particle with charge in ratio,` 1:1:2` and mass in ratio `1 : 2:4` enters in a region of uniform magnetic field. Calculate the ratio of radii of circular path described by three particles when inetic energy of all three particles is same.

Answer

To solve the problem of finding the ratio of the radii of the circular paths described by three charged particles in a uniform magnetic field, we can follow these steps:

### Step 1: Define the Charges and Masses
Let the charges of the three particles be:
- Charge of particle 1, \( q_1 = q \)
- Charge of particle 2, \( q_2 = q \)
- Charge of particle 3, \( q_3 = 2q \)

Let the masses of the three particles be:
- Mass of particle 1, \( m_1 = m \)
- Mass of particle 2, \( m_2 = 2m \)
- Mass of particle 3, \( m_3 = 4m \)

### Step 2: Use the Kinetic Energy Condition
We know that the kinetic energy \( K \) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
Since the kinetic energy of all three particles is the same, we can write:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_3 v_3^2 \]
This simplifies to:
\[ m_1 v_1^2 = m_2 v_2^2 = m_3 v_3^2 \]

### Step 3: Set Up the Ratios
Substituting the masses into the equation:
\[ m v_1^2 = 2m v_2^2 = 4m v_3^2 \]
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ v_1^2 = 2 v_2^2 = 4 v_3^2 \]

### Step 4: Express Velocities in Terms of Each Other
From the above equations, we can express the velocities:
- From \( v_1^2 = 2 v_2^2 \), we get \( v_1 = \sqrt{2} v_2 \)
- From \( v_2^2 = \frac{1}{2} v_1^2 \), we get \( v_2 = \frac{1}{\sqrt{2}} v_1 \)
- From \( v_3^2 = \frac{1}{4} v_1^2 \), we get \( v_3 = \frac{1}{2} v_1 \)

### Step 5: Calculate the Radii of the Circular Paths
The radius \( R \) of the circular path for a charged particle in a magnetic field is given by:
\[ R = \frac{mv}{qB} \]
Now, we can find the radii for each particle:
- For particle 1:
\[ R_1 = \frac{m v_1}{q B} \]
- For particle 2:
\[ R_2 = \frac{m_2 v_2}{q_2 B} = \frac{2m \cdot \frac{1}{\sqrt{2} v_1}}{q B} = \frac{2m \cdot \frac{1}{\sqrt{2} v_1}}{q B} = \frac{2m}{qB \sqrt{2}} v_1 \]
- For particle 3:
\[ R_3 = \frac{m_3 v_3}{q_3 B} = \frac{4m \cdot \frac{1}{2} v_1}{2q B} = \frac{4m}{2qB} v_1 = \frac{2m}{qB} v_1 \]

### Step 6: Find the Ratios of the Radii
Now we can find the ratios:
\[ R_1 : R_2 : R_3 = \frac{m v_1}{qB} : \frac{2m}{qB \sqrt{2}} v_1 : \frac{2m}{qB} v_1 \]
This simplifies to:
\[ R_1 : R_2 : R_3 = 1 : \frac{2}{\sqrt{2}} : 2 = 1 : \sqrt{2} : 2 \]

### Final Answer
Thus, the ratio of the radii of the circular paths described by the three particles is:
\[ R_1 : R_2 : R_3 = 1 : \sqrt{2} : 2 \]