The Speed of an electron having a wavelength of 3.0 cm is

Options

A. 2.43`ms^(−1)`
B. 0.0242`ms^(−1)`
C. 24.3`ms^(−1)`
D. 0.243`ms^(−1)`

Answer

To find the speed of an electron having a wavelength of 3.0 cm, we can use the de Broglie wavelength formula. Here’s the step-by-step solution:

### Step 1: Convert Wavelength to Meters
The given wavelength is 3.0 cm. We need to convert this into meters since the standard unit for wavelength in physics is meters.

\[ \text{Wavelength} (\lambda) = 3.0 \, \text{cm} = 3.0 \times 10^{-2} \, \text{m} = 0.03 \, \text{m} \]

### Step 2: Write Down the de Broglie Wavelength Formula
According to de Broglie, the wavelength (\(\lambda\)) of a particle is given by the formula:

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

Where:
- \(h\) = Planck's constant = \(6.626 \times 10^{-34} \, \text{J s}\)
- \(m\) = mass of the electron = \(9.1 \times 10^{-31} \, \text{kg}\)
- \(v\) = speed of the electron (which we need to find)

### Step 3: Rearrange the Formula to Solve for Speed
We need to rearrange the formula to solve for the speed (\(v\)) of the electron:

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

### Step 4: Substitute the Known Values into the Equation
Now, substitute the values of \(h\), \(m\), and \(\lambda\) into the equation:

\[ v = \frac{6.626 \times 10^{-34} \, \text{J s}}{(9.1 \times 10^{-31} \, \text{kg})(0.03 \, \text{m})} \]

### Step 5: Calculate the Speed
Now, calculate the speed:

\[ v = \frac{6.626 \times 10^{-34}}{(9.1 \times 10^{-31} \times 0.03)} \]

Calculating the denominator:

\[ 9.1 \times 10^{-31} \times 0.03 = 2.73 \times 10^{-32} \]

Now substituting back:

\[ v = \frac{6.626 \times 10^{-34}}{2.73 \times 10^{-32}} \approx 0.0242 \, \text{m/s} \]

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

\[ \boxed{0.0242 \, \text{m/s}} \]

---