A particle having a mass of 10 milligram has a velocity of 3600 km/hour. Calculate the wavelength of the particle `(h = 6.626 xx 10^(-27) "erg second")`

Options

A. `6.626 xx 10^(-30) cm`
B. `6.626 xx 10^(-31) cm`
C. `6.626 xx 10^(-28) cm`
D. `6.626 xx 10^(-29) cm`

Explanation

To solve the problem of calculating the wavelength of a particle with a mass of 10 milligrams and a velocity of 3600 km/hour, we will use the de Broglie wavelength formula:

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

where:
- \( \lambda \) is the wavelength,
- \( h \) is Planck's constant,
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

### Step 1: Convert the mass from milligrams to kilograms
The mass of the particle is given as 10 milligrams. To convert this to kilograms:

\[ 10 \text{ mg} = 10 \times 10^{-3} \text{ g} = 10 \times 10^{-6} \text{ kg} = 1 \times 10^{-5} \text{ kg} \]

### Step 2: Convert the velocity from km/hour to m/s
The velocity is given as 3600 km/hour. To convert this to meters per second:

\[ 3600 \text{ km/hour} = 3600 \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 1000 \text{ m/s} \]

### Step 3: Substitute the values into the de Broglie equation
Now that we have the mass and velocity in the correct units, we can substitute these values into the de Broglie wavelength formula. The value of Planck's constant \( h \) is given as \( 6.626 \times 10^{-27} \text{ erg s} \).

Using the conversion \( 1 \text{ erg} = 10^{-7} \text{ Joules} \), we convert \( h \) to Joules:

\[ h = 6.626 \times 10^{-27} \text{ erg s} = 6.626 \times 10^{-27} \times 10^{-7} \text{ J s} = 6.626 \times 10^{-34} \text{ J s} \]

Now we can substitute \( h \), \( m \), and \( v \) into the formula:

\[ \lambda = \frac{6.626 \times 10^{-34} \text{ J s}}{(1 \times 10^{-5} \text{ kg})(1000 \text{ m/s})} \]

### Step 4: Calculate the wavelength
Calculating the denominator:

\[ 1 \times 10^{-5} \text{ kg} \times 1000 \text{ m/s} = 1 \times 10^{-2} \text{ kg m/s} \]

Now substituting back into the equation for \( \lambda \):

\[ \lambda = \frac{6.626 \times 10^{-34}}{1 \times 10^{-2}} = 6.626 \times 10^{-32} \text{ m} \]

### Step 5: Convert the wavelength from meters to centimeters
Since the answer is required in centimeters, we convert meters to centimeters:

\[ \lambda = 6.626 \times 10^{-32} \text{ m} = 6.626 \times 10^{-30} \text{ cm} \]

### Final Answer
Thus, the wavelength of the particle is:

\[ \lambda = 6.626 \times 10^{-30} \text{ cm} \]