An `e^-` has magnetic quantum number as –3, what is its principal quantum number?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the principal quantum number (n) of an electron given its magnetic quantum number (m) as -3, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnetic Quantum Number (m)**:
- The magnetic quantum number (m) can take values from -l to +l, where l is the azimuthal quantum number (also known as the angular momentum quantum number).

2. **Identify the Maximum Value of l**:
- Since m is given as -3, we can conclude that the maximum value of l must be 3. This is because m ranges from -l to +l, and if m = -3, then l must be at least 3 to accommodate this value.

3. **Relate l to the Principal Quantum Number (n)**:
- The azimuthal quantum number l can take values from 0 to n-1. Therefore, if l is 3, we can set up the inequality:
\[ l \leq n - 1 \]
- Substituting l = 3 into the inequality gives:
\[ 3 \leq n - 1 \]

4. **Solve for n**:
- Rearranging the inequality:
\[ n \geq 3 + 1 \]
\[ n \geq 4 \]
- Since n must be a positive integer, the smallest value for n that satisfies this inequality is 4.

5. **Conclusion**:
- Therefore, the principal quantum number (n) for the electron with a magnetic quantum number of -3 is 4.

### Final Answer:
The principal quantum number (n) is **4**.

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