A particle of mass m and having charge q is moving in a circular path of radius r in the presence of a uniform magnetic field. The particle breaks into smaller fragments X and Y of equal mass. The fragment X now has charge `q/3` and the fragment Y has charge `(2q)/3`. Immediately after the fragmentation, X comes to rest. The radius of the path of Y is now `r_(Y)`. The ratio `(r_(Y))/r` is ______________.

Explanation

To solve the problem step by step, we will analyze the motion of the charged particle before and after fragmentation and apply the principles of conservation of momentum and the formula for the radius of a charged particle moving in a magnetic field.

### Step 1: Understand the initial conditions
Initially, we have a particle of mass \( m \) and charge \( q \) moving in a circular path of radius \( r \) in a uniform magnetic field \( B \). The centripetal force acting on the particle is provided by the magnetic force.

### Step 2: Write the expression for the radius of the circular path
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:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step 3: Analyze the fragmentation
After fragmentation, the particle breaks into two fragments \( X \) and \( Y \) of equal mass \( \frac{m}{2} \). Fragment \( X \) has a charge of \( \frac{q}{3} \) and comes to rest, meaning its velocity \( v_X = 0 \). Fragment \( Y \) has a charge of \( \frac{2q}{3} \).

### Step 4: Use conservation of momentum
Before fragmentation, the momentum of the system is:
\[ p_{\text{initial}} = mv \]
After fragmentation, the momentum of fragment \( X \) is \( 0 \) (since it comes to rest), and the momentum of fragment \( Y \) is:
\[ p_Y = \frac{m}{2} v_Y \]
By conservation of momentum:
\[ mv = 0 + \frac{m}{2} v_Y \]
This simplifies to:
\[ v_Y = 2v \]

### Step 5: Calculate the radius of fragment \( Y \)
Now, we can find the radius \( r_Y \) for fragment \( Y \) using the formula for the radius of a charged particle:
\[ r_Y = \frac{\left(\frac{m}{2}\right) v_Y}{\left(\frac{2q}{3}\right) B} \]
Substituting \( v_Y = 2v \):
\[ r_Y = \frac{\left(\frac{m}{2}\right)(2v)}{\left(\frac{2q}{3}\right) B} \]
This simplifies to:
\[ r_Y = \frac{mv}{\frac{2q}{3} B} = \frac{3mv}{2qB} \]

### Step 6: Find the ratio \( \frac{r_Y}{r} \)
Now we can find the ratio of the radius of fragment \( Y \) to the original radius \( r \):
\[ \frac{r_Y}{r} = \frac{\frac{3mv}{2qB}}{\frac{mv}{qB}} = \frac{3}{2} \]

### Final Answer
Thus, the ratio \( \frac{r_Y}{r} \) is:
\[ \frac{r_Y}{r} = \frac{3}{2} \]