An electron moving with kinetic energy `6xx10^(-16)` joules enters a field of magnetic induction `6xx10^(-3)" weber//m^2` at right angle to its motion . The radius of its path is

Options

A. 3.44 cm
B. 4.23 cm
C. 5.17 cm
D. 7.7 cm

Explanation

To solve the problem of finding the radius of the path of an electron moving in a magnetic field, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and velocity
The kinetic energy (KE) of an electron is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron.

### Step 2: Rearrange the kinetic energy formula to find velocity
We can rearrange the formula to solve for \( v \):
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 3: Substitute the known values
We know:
- \( KE = 6 \times 10^{-16} \) joules,
- \( m = 9.1 \times 10^{-31} \) kg (mass of the electron).

Substituting these values into the equation:
\[ v = \sqrt{\frac{2 \cdot (6 \times 10^{-16})}{9.1 \times 10^{-31}}} \]

### Step 4: Calculate the velocity
Calculating the value inside the square root:
\[ v = \sqrt{\frac{12 \times 10^{-16}}{9.1 \times 10^{-31}}} \]
\[ v = \sqrt{1.31868 \times 10^{15}} \approx 1.15 \times 10^7 \text{ m/s} \]

### Step 5: Use the formula for the radius of the circular path
The radius \( r \) of the circular path of a charged particle in a magnetic field is given by:
\[ r = \frac{mv}{qB} \]
where:
- \( q \) is the charge of the electron (\( q = 1.6 \times 10^{-19} \) coulombs),
- \( B \) is the magnetic field strength (\( B = 6 \times 10^{-3} \) weber/m²).

### Step 6: Substitute the values into the radius formula
Now substituting the values we have:
\[ r = \frac{(9.1 \times 10^{-31}) \cdot (1.15 \times 10^7)}{(1.6 \times 10^{-19}) \cdot (6 \times 10^{-3})} \]

### Step 7: Calculate the radius
Calculating the numerator:
\[ 9.1 \times 10^{-31} \cdot 1.15 \times 10^7 \approx 1.0465 \times 10^{-23} \]
Calculating the denominator:
\[ (1.6 \times 10^{-19}) \cdot (6 \times 10^{-3}) = 9.6 \times 10^{-22} \]
Now, dividing the two results:
\[ r = \frac{1.0465 \times 10^{-23}}{9.6 \times 10^{-22}} \approx 0.1094 \text{ m} \]

### Step 8: Convert to centimeters
To convert meters to centimeters, multiply by 100:
\[ r \approx 10.94 \text{ cm} \]

### Final Result
Thus, the radius of the path of the electron is approximately \( 10.94 \) cm.