The magnitude of an orbital angular momentum vector of an electron is `sqrt(6) (h)/(2pi)` into how many components will the vector split if an external field is applied to it ?

Options

A. 3
B. 5
C. 7
D. 10

Explanation

To solve the problem, we need to determine how many components the orbital angular momentum vector of an electron will split into when an external magnetic field is applied. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
The magnitude of the orbital angular momentum vector (L) is given as:
\[ L = \sqrt{6} \frac{h}{2\pi} \]

### Step 2: Use the Formula for Orbital Angular Momentum
The formula for the magnitude of the orbital angular momentum is given by:
\[ L = \sqrt{l(l + 1)} \frac{h}{2\pi} \]
where \( l \) is the azimuthal quantum number.

### Step 3: Set Up the Equation
We can equate the two expressions for \( L \):
\[ \sqrt{l(l + 1)} \frac{h}{2\pi} = \sqrt{6} \frac{h}{2\pi} \]

### Step 4: Simplify the Equation
We can cancel \( \frac{h}{2\pi} \) from both sides:
\[ \sqrt{l(l + 1)} = \sqrt{6} \]

### Step 5: Square Both Sides
Squaring both sides gives:
\[ l(l + 1) = 6 \]

### Step 6: Solve for \( l \)
This is a quadratic equation:
\[ l^2 + l - 6 = 0 \]
Factoring gives:
\[ (l - 2)(l + 3) = 0 \]
Thus, \( l = 2 \) (since \( l \) cannot be negative).

### Step 7: Determine the Orbital Type
The value \( l = 2 \) corresponds to the d-orbitals.

### Step 8: Determine the Number of Components
When an external magnetic field is applied, the d-orbitals will split into \( 2l + 1 \) components. For \( l = 2 \):
\[ 2l + 1 = 2(2) + 1 = 5 \]

### Conclusion
Therefore, the orbital angular momentum vector will split into **5 components** when an external magnetic field is applied.

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