Magnetic force acting on the charged particle projected perpendicular to magnetic field is proportional to `v^(n)`. Where v is the speed. Find n

Answer

To find the value of \( n \) in the expression for the magnetic force acting on a charged particle projected perpendicular to a magnetic field, we can follow these steps:

### Step 1: Understand the formula for magnetic force
The magnetic force \( F \) acting on a charged particle moving in a magnetic field is given by the formula:
\[ F = q v B \sin \theta \]
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 condition
In the problem, it is stated that the charged particle is projected perpendicular to the magnetic field. This means that the angle \( \theta \) is \( 90^\circ \).

### Step 3: Substitute the angle into the formula
Since \( \theta = 90^\circ \), we can substitute this into the formula:
\[ F = q v B \sin 90^\circ \]
Knowing that \( \sin 90^\circ = 1 \), we simplify the equation:
\[ F = q v B \cdot 1 \]
Thus, we have:
\[ F = q v B \]

### Step 4: Identify the proportional relationship
From the equation \( F = q v B \), we can see that the magnetic force \( F \) is directly proportional to the velocity \( v \):
\[ F \propto v \]
This implies that the force is proportional to \( v^1 \).

### Step 5: Conclude the value of \( n \)
From the proportionality \( F \propto v^1 \), we can conclude that:
\[ n = 1 \]

### Final Answer
Thus, the value of \( n \) is \( 1 \).

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