An electron is moving on a circular path of radius `r` with speed `v` in a transverse magnetic field B. e/m for it will be

Options

A. `v/Br`
B. `B/rv`
C. Brv
D. `vr/B`

Answer

To find the ratio of charge to mass (e/m) for an electron moving in a circular path in a transverse magnetic field, we can follow these steps:

### Step 1: Understand the forces acting on the electron
When an electron moves in a magnetic field, it experiences a magnetic force that acts as the centripetal force required to keep it moving in a circular path. The magnetic force \( F \) acting on the electron can be expressed as:
\[ F = Q \cdot v \cdot B \]
where:
- \( Q \) is the charge of the electron (denoted as \( e \)),
- \( v \) is the speed of the electron,
- \( B \) is the magnetic field strength.

### Step 2: Relate magnetic force to centripetal force
The centripetal force \( F_c \) required to keep the electron moving in a circular path of radius \( r \) is given by:
\[ F_c = \frac{m v^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( r \) is the radius of the circular path.

### Step 3: Set the magnetic force equal to the centripetal force
Since the magnetic force provides the necessary centripetal force, we can set these two expressions equal to each other:
\[ Q \cdot v \cdot B = \frac{m v^2}{r} \]

### Step 4: Substitute the charge of the electron
Substituting \( Q \) with \( e \) (the charge of the electron), we have:
\[ e \cdot v \cdot B = \frac{m v^2}{r} \]

### Step 5: Rearrange the equation to find e/m
To find the ratio \( \frac{e}{m} \), we can rearrange the equation:
\[ e = \frac{m v}{r} \cdot B \]
Dividing both sides by \( m \):
\[ \frac{e}{m} = \frac{v \cdot B}{r} \]

### Conclusion
Thus, the ratio of charge to mass for the electron is:
\[ \frac{e}{m} = \frac{v \cdot B}{r} \]

### Final Answer
The correct option for the value of \( \frac{e}{m} \) is:
\[ \frac{e}{m} = \frac{B \cdot v}{r} \]

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