A hydrogen ion moving with a velocity of `10^(4)m//s` enters in magentic field `10^(-4)T` perpendicularly, then its radius of circular path is

Options

A. 10.01 m
B. 1.01 m
C. 101 m
D. 0.101 m

Answer

To find the radius of the circular path of a hydrogen ion moving in a magnetic field, we can use the relationship between the magnetic force acting on the ion and the centripetal force required to keep it in circular motion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Velocity of the hydrogen ion, \( v = 10^4 \, \text{m/s} \)
- Magnetic field strength, \( B = 10^{-4} \, \text{T} \)
- Charge of a hydrogen ion, \( q = 1.6 \times 10^{-19} \, \text{C} \)
- Mass of a hydrogen ion, \( m = 1.67 \times 10^{-27} \, \text{kg} \)

2. **Set Up the Equations:**
The magnetic force \( F_m \) acting on the ion is given by:
\[ F_m = q v B \]
The centripetal force \( F_c \) required to keep the ion in circular motion is given by:
\[ F_c = \frac{m v^2}{r} \]
Where \( r \) is the radius of the circular path.

3. **Equate the Forces:**
Since the magnetic force provides the centripetal force, we can set these two equations equal to each other:
\[ q v B = \frac{m v^2}{r} \]

4. **Rearrange to Solve for \( r \):**
Rearranging the equation to find the radius \( r \):
\[ r = \frac{m v}{q B} \]

5. **Substitute the Known Values:**
Now, substitute the values into the equation:
\[ r = \frac{(1.67 \times 10^{-27} \, \text{kg}) (10^4 \, \text{m/s})}{(1.6 \times 10^{-19} \, \text{C}) (10^{-4} \, \text{T})} \]

6. **Calculate the Numerator:**
\[ \text{Numerator} = 1.67 \times 10^{-27} \times 10^4 = 1.67 \times 10^{-23} \]

7. **Calculate the Denominator:**
\[ \text{Denominator} = 1.6 \times 10^{-19} \times 10^{-4} = 1.6 \times 10^{-23} \]

8. **Final Calculation:**
Now, divide the numerator by the denominator:
\[ r = \frac{1.67 \times 10^{-23}}{1.6 \times 10^{-23}} \approx 1.04 \, \text{m} \]

### Conclusion:
The radius of the circular path of the hydrogen ion in the magnetic field is approximately \( r \approx 1.04 \, \text{m} \).