Two particles A and B having equal charges `+6C`, after being accelerated through the same potential differences, enter a region of uniform magnetic field and describe circular paths of radii `2cm` and `3cm` respectively. The ratio of mass of A to that of B is

Options

A. `4//9`
B. `9//5`
C. `1//2`
D. `1//3`

Answer

To solve the problem of finding the ratio of the masses of two charged particles A and B, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and potential difference
When a charged particle is accelerated through a potential difference (V), it gains kinetic energy. The kinetic energy (KE) gained by the particle is given by the equation:
\[ \text{KE} = \frac{1}{2} mv^2 = qV \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle after acceleration,
- \( q \) is the charge of the particle,
- \( V \) is the potential difference.

### Step 2: Relate the radius of the circular path to mass and velocity
When a charged particle enters a magnetic field, it moves in a circular path. The centripetal force required for circular motion is provided by the magnetic force. The equation for this is:
\[ \frac{mv^2}{r} = qvB \]
where:
- \( r \) is the radius of the circular path,
- \( B \) is the magnetic field strength.

### Step 3: Solve for velocity (v)
From the centripetal force equation, we can rearrange it to find the velocity:
\[ mv^2 = qvBr \implies v = \frac{qBr}{m} \]

### Step 4: Substitute velocity into the kinetic energy equation
Now, we can substitute this expression for \( v \) back into the kinetic energy equation:
\[ \frac{1}{2} m \left(\frac{qBr}{m}\right)^2 = qV \]
This simplifies to:
\[ \frac{1}{2} \frac{q^2 B^2 r^2}{m} = qV \]
Rearranging gives:
\[ r^2 = \frac{2mV}{q^2B^2} \]

### Step 5: Establish the relationship between radius and mass
From the equation \( r^2 = \frac{2mV}{q^2B^2} \), we can see that \( r^2 \) is directly proportional to \( m \):
\[ r^2 \propto m \]
Thus, we can set up the ratio of the radii and masses for particles A and B:
\[ \frac{r_A^2}{r_B^2} = \frac{m_A}{m_B} \]

### Step 6: Substitute the given values
We know:
- \( r_A = 2 \, \text{cm} = 0.02 \, \text{m} \)
- \( r_B = 3 \, \text{cm} = 0.03 \, \text{m} \)

Calculating the squares:
\[ r_A^2 = (0.02)^2 = 0.0004 \, \text{m}^2 \]
\[ r_B^2 = (0.03)^2 = 0.0009 \, \text{m}^2 \]

### Step 7: Find the ratio of masses
Now we can find the ratio of the masses:
\[ \frac{m_A}{m_B} = \frac{r_A^2}{r_B^2} = \frac{0.0004}{0.0009} = \frac{4}{9} \]

### Final Answer
Thus, the ratio of the mass of particle A to the mass of particle B is:
\[ \frac{m_A}{m_B} = \frac{4}{9} \]

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