A charged particle is moving in a circular orbit of radius 6 cm with a uniform speed of `3 xxx 10^(6)` m/s under the action of a uniform magnetic field `2 xx 10^(-4) wb//m^(2)` at right angles to the plane of the orbit. The charge to mass ratio of the particle is

Options

A. `5 xx 10^(9)` C /kg
B. `2.5 xx 10^(11)` C/kg
C. `5 xx 10^(11)` C /kg
D. `5 xx 10^(12)` C/kg

Explanation

To find the charge to mass ratio (Q/M) of a charged particle moving in a circular orbit under the influence of a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Radius of the orbit (r) = 6 cm = 6 × 10^(-2) m
- Speed of the particle (v) = 3 × 10^(6) m/s
- Magnetic field (B) = 2 × 10^(-4) Wb/m²

2. **Use the Formula for Radius of Circular Motion in a Magnetic Field**:
The radius of the circular motion of a charged particle in a magnetic field is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- m = mass of the particle
- v = velocity of the particle
- q = charge of the particle
- B = magnetic field strength

3. **Rearranging the Formula to Find Charge to Mass Ratio**:
We need to find the charge to mass ratio (Q/M). Rearranging the formula gives:
\[ \frac{q}{m} = \frac{v}{rB} \]

4. **Substituting the Known Values**:
Now, substituting the values we have:
\[ \frac{q}{m} = \frac{3 \times 10^{6}}{(6 \times 10^{-2})(2 \times 10^{-4})} \]

5. **Calculating the Denominator**:
First, calculate the denominator:
\[ (6 \times 10^{-2})(2 \times 10^{-4}) = 12 \times 10^{-6} = 1.2 \times 10^{-5} \]

6. **Calculating Q/M**:
Now substitute this back into the equation:
\[ \frac{q}{m} = \frac{3 \times 10^{6}}{1.2 \times 10^{-5}} = \frac{3}{1.2} \times 10^{6 + 5} = 2.5 \times 10^{11} \]

7. **Final Result**:
Therefore, the charge to mass ratio of the particle is:
\[ \frac{q}{m} = 2.5 \times 10^{11} \, \text{C/kg} \]