The Speed of an electron having a wavelength of 4.0 cm is

Options

A. 0.0181`ms^(−1)`
B. 1.18`ms^(−1)`
C. 11.81`ms^(−1)`
D. 0.118`ms^(−1)`

Answer

To find the speed of an electron with a wavelength of 4.0 cm, we can use the de Broglie wavelength formula. Here’s a step-by-step solution:

### Step 1: Convert Wavelength to Meters
The given wavelength (λ) is 4.0 cm. We need to convert this to meters.

\[ \lambda = 4.0 \, \text{cm} = 4.0 \times 10^{-2} \, \text{m} = 0.04 \, \text{m} \]

### Step 2: Write Down the De Broglie Wavelength Formula
According to de Broglie's equation, the wavelength (λ) of a particle is given by:

\[ \lambda = \frac{h}{mv} \]

Where:
- \( h \) = Planck's constant = \( 6.626 \times 10^{-34} \, \text{J s} \)
- \( m \) = mass of the electron = \( 9.11 \times 10^{-31} \, \text{kg} \)
- \( v \) = velocity of the electron

### Step 3: Rearrange the Formula to Solve for Velocity
We need to find the velocity (v) of the electron. Rearranging the formula gives:

\[ v = \frac{h}{m\lambda} \]

### Step 4: Substitute the Values into the Formula
Now, substitute the known values into the equation:

\[ v = \frac{6.626 \times 10^{-34} \, \text{J s}}{(9.11 \times 10^{-31} \, \text{kg})(0.04 \, \text{m})} \]

### Step 5: Calculate the Velocity
Now, calculate the value:

\[ v = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31})(0.04)} \]

Calculating the denominator:

\[ 9.11 \times 10^{-31} \times 0.04 = 3.644 \times 10^{-32} \]

Now, substituting back:

\[ v = \frac{6.626 \times 10^{-34}}{3.644 \times 10^{-32}} \approx 0.0182 \, \text{m/s} \]

### Final Answer
The speed of the electron is approximately:

\[ v \approx 0.0182 \, \text{m/s} \]