In a region of magnetic field 0.05 T directed along positive direction of Y-axis, a beam of charged particles enters the region along X-axis with a velocity of `2.5 xx 10^(4) m s^(-1)`. Calculate the radius of circular path described by the charge particles if their charge to mass ratio is `7.5 xx 10^(6) C kg^(-1)`.

Answer

To solve the problem of finding the radius of the circular path described by charged particles in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Magnetic field strength, \( B = 0.05 \, \text{T} \)
- Velocity of the charged particles, \( v = 2.5 \times 10^4 \, \text{m/s} \)
- Charge to mass ratio, \( \frac{Q}{m} = 7.5 \times 10^6 \, \text{C/kg} \)

2. **Use the Formula for Radius of Circular Motion in a Magnetic Field**:
The radius \( r \) of the circular path of a charged particle moving in a magnetic field is given by the formula:
\[ r = \frac{mv}{qB} \]
However, we can express this in terms of the charge to mass ratio:
\[ r = \frac{v}{\frac{Q}{m} \cdot B} \]

3. **Substitute the Values into the Formula**:
Now we can substitute the known values into the formula:
\[ r = \frac{2.5 \times 10^4}{7.5 \times 10^6 \cdot 0.05} \]

4. **Calculate the Denominator**:
First, calculate the denominator:
\[ 7.5 \times 10^6 \cdot 0.05 = 3.75 \times 10^5 \]

5. **Calculate the Radius**:
Now substitute this value back into the equation for \( r \):
\[ r = \frac{2.5 \times 10^4}{3.75 \times 10^5} \]
Simplifying this gives:
\[ r = \frac{2.5}{3.75} \times 10^{-1} = \frac{2.5}{3.75} \times 0.1 \]
\[ r = \frac{2.5}{3.75} \times 0.1 = \frac{1}{1.5} \times 0.1 \approx 0.06667 \, \text{m} \]

6. **Convert to Centimeters**:
To express the radius in centimeters:
\[ r \approx 6.67 \, \text{cm} \]

### Final Answer:
The radius of the circular path described by the charged particles is approximately \( 6.67 \, \text{cm} \).

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