Which of the follwing particles will have minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field?

Options

A. Electron
B. Proton
C. `He^(+)`
D. `Li^(++)`

Answer

To determine which particle will have the minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field, we can follow these steps:

### Step 1: Understand the relationship between frequency, charge, and mass
The frequency of revolution \( f \) of a charged particle in a magnetic field is given by the formula:
\[ f = \frac{Q B}{2 \pi m} \]
where:
- \( Q \) is the charge of the particle,
- \( B \) is the magnetic field strength,
- \( m \) is the mass of the particle.

### Step 2: Identify the factors affecting frequency
From the formula, we can see that the frequency \( f \) is directly proportional to the charge \( Q \) and the magnetic field \( B \), and inversely proportional to the mass \( m \). Therefore, to minimize the frequency, we need to maximize the mass \( m \).

### Step 3: List the particles and their masses
Now, let's consider the particles given in the options:
- **Electron**: Mass \( m_e = 9.1 \times 10^{-31} \) kg
- **Proton**: Mass \( m_p = 1.67 \times 10^{-27} \) kg
- **Helium ion (He\(^+\))**: Mass \( m_{He} = 4 \times m_p = 4 \times 1.67 \times 10^{-27} \) kg \( = 6.68 \times 10^{-27} \) kg
- **Lithium ion (Li\(^+\))**: Mass \( m_{Li} = 7 \times m_p = 7 \times 1.67 \times 10^{-27} \) kg \( = 1.17 \times 10^{-26} \) kg

### Step 4: Compare the masses
Now we can compare the masses:
- Mass of Electron: \( 9.1 \times 10^{-31} \) kg
- Mass of Proton: \( 1.67 \times 10^{-27} \) kg
- Mass of Helium ion: \( 6.68 \times 10^{-27} \) kg
- Mass of Lithium ion: \( 1.17 \times 10^{-26} \) kg

### Step 5: Determine which particle has the maximum mass
From the comparison, we can see that:
- Lithium ion has the maximum mass.

### Step 6: Conclusion
Since the frequency is inversely proportional to the mass, the particle with the maximum mass (Lithium ion) will have the minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field.

Thus, the answer is:
**Lithium ion (Li\(^+\)) will have the minimum frequency of revolution.**

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