The number of waves in the fourth bohr orbit of hydrogen is

Options

A. 3
B. 4
C. 9
D. 12

Explanation

To find the number of waves in the fourth Bohr orbit of hydrogen, we can follow these steps:

### Step 1: Understand the relationship between the circumference of the orbit and the wavelength
The electron in the Bohr orbit travels in a circular path, and the distance it travels in one complete revolution is equal to the circumference of the orbit. The circumference \( C \) of the orbit is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the orbit.

### Step 2: Relate the circumference to the wavelength and number of waves
The number of waves \( n \) in the orbit can be expressed as:
\[ C = n \cdot \lambda \]
where \( \lambda \) is the wavelength of the electron.

### Step 3: Substitute the expression for wavelength
According to de Broglie's hypothesis, the wavelength \( \lambda \) of the electron can be expressed as:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is its velocity.

### Step 4: Combine the equations
Substituting the expression for \( \lambda \) into the circumference equation gives us:
\[ 2\pi r = n \cdot \frac{h}{mv} \]

### Step 5: Rearranging the equation to find \( n \)
Rearranging the above equation to solve for \( n \):
\[ n = \frac{2\pi r mv}{h} \]

### Step 6: Use the angular momentum quantization condition
According to Bohr's model, the angular momentum \( L \) of the electron in the nth orbit is quantized and given by:
\[ L = mv r = n \cdot \frac{h}{2\pi} \]
For the fourth orbit (\( n = 4 \)):
\[ mv r = 4 \cdot \frac{h}{2\pi} \]

### Step 7: Substitute the angular momentum into the equation for \( n \)
Now substituting \( mv r \) into the equation for \( n \):
\[ n = \frac{2\pi r \cdot \left(4 \cdot \frac{h}{2\pi}\right)}{h} \]

### Step 8: Simplifying the expression
This simplifies to:
\[ n = \frac{2\pi r \cdot 4 \cdot \frac{h}{2\pi}}{h} = 4 \]

### Conclusion
Thus, the number of waves in the fourth Bohr orbit of hydrogen is:
\[ \boxed{4} \]