The ratio of magnetic dipole moment to angular momentum of electron is

Options

A. `e/m`
B. `(2e)/m`
C. `e/(2m)`
D. `e/4m`

Answer

To find the ratio of the magnetic dipole moment (μ) to the angular momentum (L) of an electron, we can follow these steps:

### Step 1: Define the magnetic dipole moment
The magnetic dipole moment (μ) of a current loop is given by the formula:
\[ \mu = I \cdot A \]
where \(I\) is the current and \(A\) is the area of the loop.

### Step 2: Calculate the current (I)
The current (I) due to the electron moving in a circular orbit can be expressed as:
\[ I = \frac{e}{T} \]
where \(e\) is the charge of the electron and \(T\) is the time period of one complete revolution. The time period \(T\) can be calculated as:
\[ T = \frac{2\pi r}{v} \]
where \(r\) is the radius of the orbit and \(v\) is the velocity of the electron. Thus, we can substitute \(T\) into the equation for current:
\[ I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

### Step 3: Calculate the area (A)
The area (A) of the circular path of the electron is given by:
\[ A = \pi r^2 \]

### Step 4: Substitute I and A into the formula for μ
Now substituting \(I\) and \(A\) into the formula for magnetic dipole moment:
\[ \mu = I \cdot A = \left(\frac{ev}{2\pi r}\right) \cdot (\pi r^2) = \frac{evr}{2} \]

### Step 5: Define angular momentum (L)
The angular momentum (L) of the electron is given by:
\[ L = mvr \]
where \(m\) is the mass of the electron.

### Step 6: Calculate the ratio of μ to L
Now we can find the ratio of the magnetic dipole moment to angular momentum:
\[ \frac{\mu}{L} = \frac{\frac{evr}{2}}{mvr} \]
Simplifying this expression:
\[ \frac{\mu}{L} = \frac{evr}{2mvr} = \frac{e}{2m} \]

### Conclusion
Thus, the ratio of the magnetic dipole moment to the angular momentum of the electron is:
\[ \frac{\mu}{L} = \frac{e}{2m} \]

### Final Answer
The ratio of magnetic dipole moment to angular momentum of the electron is \(\frac{e}{2m}\).

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