The Speed of an electron having a wavelength of 1.0 cm is

Options

A. 0.0108`ms^(-1)`
B. 0.0727`ms^(-1)`
C. 7.27`ms^(-1)`
D. 72.78`ms^(-1)`

Answer

To find the speed of an electron with a wavelength of 1.0 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 1.0 cm. We need to convert this into meters for consistency in SI units.

\[ \lambda = 1.0 \, \text{cm} = 1.0 \times 10^{-2} \, \text{m} = 0.01 \, \text{m} \]

**Hint:** Remember that 1 cm = 0.01 m.

### Step 2: Use the de Broglie Wavelength Formula
The de Broglie wavelength formula relates the wavelength (λ) of a particle to its momentum (p):

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

Where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \))
- \( m \) is the mass of the electron (\( 9.1 \times 10^{-31} \, \text{kg} \))
- \( v \) is the speed of the electron

### Step 3: Rearranging the Formula
We need to find the speed (v) of the electron, so we rearrange the formula to solve for \( v \):

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

### Step 4: Substitute the Values
Now we can substitute the known values into the rearranged formula:

\[ v = \frac{6.626 \times 10^{-34} \, \text{Js}}{(9.1 \times 10^{-31} \, \text{kg})(0.01 \, \text{m})} \]

### Step 5: Calculate the Speed
Now we perform the calculation:

\[ v = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-33}} \approx 0.0727 \, \text{m/s} \]

### Final Answer
The speed of the electron having a wavelength of 1.0 cm is approximately:

\[ v \approx 0.0727 \, \text{m/s} \]

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