An electron moves along a circular path of radius 10 cm. If the centripetal acceleration is `4xx10^(11) m//s^(2)` , then its linear speed is

Options

A. `0.5xx10^(5) m//s`
B. `10^(5) m//s`
C. `2xx10^(5)m//s`
D. `3xx10^(5)m//s`

Explanation

To solve the problem of finding the linear speed of an electron moving along a circular path with a given centripetal acceleration, we can follow these steps:

### Step 1: Write down the given data
- Radius of the circular path, \( r = 10 \, \text{cm} \)
- Centripetal acceleration, \( a_r = 4 \times 10^{11} \, \text{m/s}^2 \)

### Step 2: Convert the radius from centimeters to meters
Since the acceleration is given in meters per second squared, we need to convert the radius from centimeters to meters:
\[ r = 10 \, \text{cm} = 10 \times 10^{-2} \, \text{m} = 0.1 \, \text{m} \]

### Step 3: Use the formula for centripetal acceleration
The formula for centripetal acceleration is given by:
\[ a_r = \frac{v^2}{r} \]
where \( v \) is the linear speed.

### Step 4: Rearrange the formula to solve for linear speed \( v \)
Rearranging the formula gives us:
\[ v^2 = a_r \times r \]

### Step 5: Substitute the known values into the equation
Now we can substitute the values of \( a_r \) and \( r \):
\[ v^2 = (4 \times 10^{11} \, \text{m/s}^2) \times (0.1 \, \text{m}) \]
\[ v^2 = 4 \times 10^{11} \times 10^{-1} = 4 \times 10^{10} \, \text{m}^2/\text{s}^2 \]

### Step 6: Calculate the linear speed \( v \)
To find \( v \), we take the square root of \( v^2 \):
\[ v = \sqrt{4 \times 10^{10}} = \sqrt{4} \times \sqrt{10^{10}} = 2 \times 10^{5} \, \text{m/s} \]

### Step 7: Conclusion
Thus, the linear speed of the electron is:
\[ v = 2 \times 10^{5} \, \text{m/s} \]

### Final Answer
The correct option from the given choices is \( 2 \times 10^{5} \, \text{m/s} \).

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