Suppose the electron in a hydrogen atom makes transition from `n = 3 to n = 2 in 10^(-8)s` The order of the torque acting on the electon in this period, using the relation between torque and angular momentum as discussed in the chapter on rotational machanics is

Options

A. `10^(-34) N m`
B. `10^(-24) N m`
C. `10^(-42) N m`
D. `10^(-8) N m`

Answer

To solve the problem, we need to determine the average torque acting on the electron during its transition from \( n = 3 \) to \( n = 2 \) in a hydrogen atom over a time interval of \( 10^{-8} \) seconds. We will use the relationship between torque and angular momentum.

### Step-by-Step Solution:

1. **Understand the relationship between torque and angular momentum**:
The average torque (\( \tau_{\text{avg}} \)) is related to the change in angular momentum (\( \Delta L \)) over time (\( \Delta t \)):
\[ \tau_{\text{avg}} = \frac{\Delta L}{\Delta t} \]

2. **Determine the initial and final angular momentum**:
The angular momentum of an electron in a hydrogen atom is given by:
\[ L = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

- For \( n_1 = 3 \):
\[ L_1 = 3 \frac{h}{2\pi} \]
- For \( n_2 = 2 \):
\[ L_2 = 2 \frac{h}{2\pi} \]

3. **Calculate the change in angular momentum (\( \Delta L \))**:
\[ \Delta L = L_2 - L_1 = \left(2 \frac{h}{2\pi}\right) - \left(3 \frac{h}{2\pi}\right) = -\frac{h}{2\pi} \]

4. **Substitute \( \Delta L \) into the torque equation**:
Now, substituting \( \Delta L \) into the average torque formula:
\[ \tau_{\text{avg}} = \frac{-\frac{h}{2\pi}}{\Delta t} \]
Given \( \Delta t = 10^{-8} \) seconds, we have:
\[ \tau_{\text{avg}} = -\frac{h}{2\pi \times 10^{-8}} \]

5. **Substitute the value of Planck's constant (\( h \))**:
The value of Planck's constant is approximately \( 6.63 \times 10^{-34} \) J·s. Thus:
\[ \tau_{\text{avg}} = -\frac{6.63 \times 10^{-34}}{2\pi \times 10^{-8}} \]

6. **Calculate the magnitude of the average torque**:
First, calculate \( 2\pi \):
\[ 2\pi \approx 6.28 \]
Now substituting this back:
\[ \tau_{\text{avg}} = -\frac{6.63 \times 10^{-34}}{6.28 \times 10^{-8}} \approx -1.055 \times 10^{-26} \text{ N·m} \]

7. **Estimate the order of magnitude**:
The order of magnitude of \( 1.055 \times 10^{-26} \) is approximately \( 10^{-24} \) N·m.

### Final Answer:
The order of the torque acting on the electron during the transition is \( 10^{-24} \) N·m.