To determine which of the four charged particles will have the minimum frequency of rotation when projected perpendicularly into a magnetic field with equal velocity, we can follow these steps:
### Step 1: Understand the relationship between frequency, charge, and mass
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. The frequency of rotation (ν) of the particle can be expressed as:
\[
ν = \frac{qB}{2πm}
\]
where:
- \(ν\) = frequency of rotation
- \(q\) = charge of the particle
- \(B\) = magnetic field strength (constant for all particles)
- \(m\) = mass of the particle
### Step 2: Analyze the formula
From the formula, we can see that the frequency of rotation 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 of the particle.
### Step 3: Identify the particles and their masses
Now, we will identify the four charged particles and their respective masses:
1. **Proton**: Mass = \(1.67 \times 10^{-27} \, \text{kg}\)
2. **Electron**: Mass = \(9.1 \times 10^{-31} \, \text{kg}\)
3. **Lithium ion (Li\(^+\))**: Mass = \(1.15 \times 10^{-26} \, \text{kg}\)
4. **Helium ion (He\(^+\))**: Mass = \(6.64 \times 10^{-27} \, \text{kg}\)
### Step 4: Compare the masses
Now, we will compare the masses of the particles:
- Proton: \(1.67 \times 10^{-27} \, \text{kg}\)
- Electron: \(9.1 \times 10^{-31} \, \text{kg}\)
- Lithium ion (Li\(^+\)): \(1.15 \times 10^{-26} \, \text{kg}\)
- Helium ion (He\(^+\)): \(6.64 \times 10^{-27} \, \text{kg}\)
### Step 5: Determine the particle with the maximum mass
From the comparison, we can see that the Lithium ion (Li\(^+\)) has the maximum mass of \(1.15 \times 10^{-26} \, \text{kg}\).
### Step 6: Conclusion
Since the frequency of rotation is inversely proportional to the mass, the particle with the maximum mass (Li\(^+\)) will have the minimum frequency of rotation.
Thus, the answer is **Lithium ion (Li\(^+\))**.
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