Which of the following particles will describe wll experience maximum magnetic force(magnitude) 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 experience the maximum magnetic force when projected with the same velocity perpendicular to a magnetic field, we can follow these steps:

### Step 1: Understand the Formula for Magnetic Force
The magnetic force \( F \) experienced by a charged particle moving in a magnetic field is given by the formula:
\[ F = q \cdot v \cdot B \cdot \sin(\theta) \]
where:
- \( F \) is the magnetic force,
- \( q \) is the charge of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the velocity vector and the magnetic field vector.

### Step 2: Analyze the Given Conditions
In this problem, we are given that:
- The velocity \( v \) is constant for all particles.
- The magnetic field \( B \) is also constant.
- The angle \( \theta \) is \( 90^\circ \) (since the particles are projected perpendicular to the magnetic field), which means \( \sin(90^\circ) = 1 \).

### Step 3: Simplify the Force Equation
Since \( v \), \( B \), and \( \sin(\theta) \) are constant, the magnetic force simplifies to:
\[ F = q \cdot k \]
where \( k = v \cdot B \) is a constant for this scenario. Therefore, the magnetic force \( F \) depends only on the charge \( q \) of the particle.

### Step 4: Compare the Charges of the Given Particles
Now, we need to compare the charges of the given particles:
1. **Electron**: Charge \( q = -1.6 \times 10^{-19} \) C
2. **Proton**: Charge \( q = +1.6 \times 10^{-19} \) C
3. **Helium Positive Ion (He\(^+\))**: Charge \( q = +2 \times 1.6 \times 10^{-19} \) C
4. **Lithium Two Positive Ion (Li\(^{2+}\))**: Charge \( q = +3 \times 1.6 \times 10^{-19} \) C

### Step 5: Determine the Maximum Charge
From the charges calculated:
- Electron: \( 1.6 \times 10^{-19} \) C
- Proton: \( 1.6 \times 10^{-19} \) C
- Helium Ion: \( 3.2 \times 10^{-19} \) C
- Lithium Ion: \( 4.8 \times 10^{-19} \) C

The Lithium Two Positive Ion (Li\(^{2+}\)) has the maximum charge.

### Step 6: Conclusion
Since the magnetic force is directly proportional to the charge, the particle with the maximum charge will experience the maximum magnetic force. Therefore, the answer is:
**Lithium Two Positive Ion (Li\(^{2+}\))**.

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