The path of a charged particle in a uniform magnetic field depends on the angle `theta` between velocity vector and magnetic field, When `theta is 0^(@) or 180^(@), F_(m) = 0` hence path of a charged particle will be linear.
When `theta = 90^(@)`, the magnetic force is perpendicular to velocity at every instant. Hence path is a circle of radius `r = (mv)/(qB)`.
The time period for circular path will be `T = (2pim)/(qB)`
When `theta` is other than `0^(@), 180^(@) and 90^(@)`, velocity can be resolved into two components, one along `vec(B)` and perpendicular to B.
`v_(|/|)=cos theta`
`v_(^)= v sin theta`
The `v_(_|_)` component gives circular path and `v_(|/|)` givestraingt line path. The resultant path is a helical path. The radius of helical path
`r=(mv sin theta)/(qB)`
ich of helix is defined as `P=v_(|/|)T`
`P=(2 i mv cos theta)`
`p=(2 pi mv cos theta)/(qB)`
Which particle will have minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field.

Options

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

Explanation

To solve the problem step by step, we need to analyze the motion of charged particles in a uniform magnetic field and derive the necessary relationships.

### Step 1: Understand the Motion of Charged Particles in a Magnetic Field
When a charged particle moves in a magnetic field, the magnetic force acting on it is given by:
\[ F_m = q(\vec{v} \times \vec{B}) \]
where \( q \) is the charge of the particle, \( \vec{v} \) is its velocity, and \( \vec{B} \) is the magnetic field.

### Step 2: Analyze Different Angles
1. **When \( \theta = 0^\circ \) or \( \theta = 180^\circ \)**: The magnetic force \( F_m = 0 \), and the particle moves in a straight line.
2. **When \( \theta = 90^\circ \)**: The magnetic force is perpendicular to the velocity, causing the particle to move in a circular path with radius:
\[ r = \frac{mv}{qB} \]
where \( m \) is the mass of the particle and \( v \) is its speed.

### Step 3: Time Period for Circular Motion
The time period \( T \) for the circular motion of the particle is given by:
\[ T = \frac{2\pi m}{qB} \]

### Step 4: General Case for Other Angles
When \( \theta \) is neither \( 0^\circ \), \( 90^\circ \), nor \( 180^\circ \), the velocity can be resolved into two components:
- Parallel to the magnetic field:
\[ v_{||} = v \cos \theta \]
- Perpendicular to the magnetic field:
\[ v_{\perp} = v \sin \theta \]

### Step 5: Helical Path
The component \( v_{\perp} \) causes circular motion, while \( v_{||} \) causes linear motion along the direction of the magnetic field. Thus, the resultant path is a helical path.

### Step 6: Radius of Helical Path
The radius of the helical path is given by:
\[ r = \frac{mv \sin \theta}{qB} \]

### Step 7: Pitch of Helix
The pitch \( P \) of the helix is defined as:
\[ P = v_{||} T = v \cos \theta \cdot \frac{2\pi m}{qB} \]
This simplifies to:
\[ P = \frac{2\pi mv \cos \theta}{qB} \]

### Step 8: Frequency of Revolution
The frequency \( f \) of revolution for a charged particle in a magnetic field is given by:
\[ f = \frac{qB}{2\pi m} \]
This indicates that the frequency is inversely proportional to the mass of the particle.

### Step 9: Determine Which Particle Has Minimum Frequency
To find which particle has the minimum frequency of revolution when projected with the same velocity perpendicular to a magnetic field, we need to consider the mass of the particles. The particle with the highest mass will have the lowest frequency.

### Conclusion
Among the given options (Li\(^+\), He\(^+\), Electron, and Proton), the Lithium ion (Li\(^+\)) has the highest mass, thus it will have the minimum frequency of revolution.