Number of waves made by a Bohr electron in one complete in its fourth orbit is

Options

A. 2
B. 3
C. 4
D. Infinite.

Explanation

To find the number of waves made by a Bohr electron in one complete revolution in its fourth orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Orbit Number (n)**:
The problem states that we are dealing with the fourth orbit, so we have:
\[ n = 4 \]

2. **Determine the Circumference of the Orbit**:
The circumference of the orbit (C) can be expressed as:
\[ C = 2\pi R_n \]
For the fourth orbit, this becomes:
\[ C = 2\pi R_4 \]

3. **Use the Bohr Model Equation**:
According to Bohr's model, the angular momentum of the electron is quantized and given by:
\[ MvR_n = \frac{nH}{2\pi} \]
For the fourth orbit (n = 4), we can write:
\[ MvR_4 = \frac{4H}{2\pi} \]

4. **Relate Velocity and Wavelength**:
The de Broglie wavelength (λ) is given by:
\[ \lambda = \frac{H}{Mv} \]
Rearranging gives us:
\[ Mv = \frac{H}{\lambda} \]

5. **Substitute Mv into the Angular Momentum Equation**:
From the angular momentum equation, we can substitute \(Mv\):
\[ \frac{H}{\lambda} R_4 = \frac{4H}{2\pi} \]

6. **Simplify the Equation**:
Cancel \(H\) from both sides:
\[ \frac{R_4}{\lambda} = \frac{4}{2\pi} \]
Rearranging gives:
\[ 2\pi R_4 = 4\lambda \]

7. **Calculate the Number of Waves**:
The number of wavelengths (N) that fit into the circumference of the orbit is given by:
\[ N = \frac{C}{\lambda} = \frac{2\pi R_4}{\lambda} \]
From our previous result, we found that:
\[ 2\pi R_4 = 4\lambda \]
Thus:
\[ N = \frac{4\lambda}{\lambda} = 4 \]

### Final Answer:
The number of waves made by a Bohr electron in one complete revolution in its fourth orbit is **4**.

---