Step 1: Understanding the Concept:
The physical phenomenon described in this question is the interaction between a moving electric charge and an external magnetic field, which results in the Magnetic Lorentz Force.
In classical electromagnetism, a stationary charge produces only an electric field.
However, once a charge starts moving, it constitutes a microscopic current, which in turn generates its own magnetic field.
When this moving charge enters an external magnetic field \( \vec{B} \), the interaction between the two magnetic fields (the one generated by the moving charge and the external one) creates a mechanical force.
This force is unique because it is always perpendicular to the direction of motion and the direction of the magnetic field lines.
This means the magnetic force can change the direction of a particle's velocity (causing it to move in a circular or helical path) but it can never change the speed of the particle, as the work done by the magnetic force is always zero.
Step 2: Key Formula or Approach:
The magnitude of the magnetic force acting on a point charge is mathematically expressed by the Lorentz force formula (specifically the magnetic component):
\[ F = q(\vec{v} \times \vec{B}) \]
In terms of magnitude, this is represented as:
\[ F = q \cdot v \cdot B \cdot \sin(\theta) \]
Where:
\( q \) is the magnitude of the electric charge in Coulombs (C).
\( v \) is the velocity of the particle in meters per second (m/s).
\( B \) is the magnetic flux density or magnetic field strength in Tesla (T).
\( \theta \) is the angle between the velocity vector and the magnetic field vector.
Step 3: Detailed Explanation:
Let us identify the given values from the problem statement:
1. The charge of the particle is \( q = 2 \) C.
2. The velocity at which the charge is moving is \( v = 3 \) m/s.
3. The external magnetic field strength is \( B = 2 \) T.
4. The direction of motion is specified as "perpendicular" to the magnetic field.
The term "perpendicular" is a critical geometric descriptor indicating that the angle \( \theta \) is exactly \( 90^{\circ} \).
In trigonometry, the sine of \( 90^{\circ} \) is equal to \( 1 \), which represents the maximum value the sine function can achieve.
This implies that the magnetic force is at its absolute maximum when a charge moves at a right angle to the field lines.
If the charge were moving parallel to the field (\( \theta = 0^{\circ} \)) or anti-parallel (\( \theta = 180^{\circ} \)), the force would be zero because \( \sin(0^{\circ}) = \sin(180^{\circ}) = 0 \).
Substituting our specific values into the formula:
\[ F = (2) \cdot (3) \cdot (2) \cdot \sin(90^{\circ}) \]
\[ F = 2 \cdot 3 \cdot 2 \cdot 1 \]
\[ F = 6 \cdot 2 \]
\[ F = 12 \text{ N} \]
The resulting force is \( 12 \) Newtons.
Furthermore, to determine the direction of this force, one would apply Fleming’s Left-Hand Rule.
By pointing the forefinger in the direction of the magnetic field (into or out of the page) and the middle finger in the direction of the positive charge's velocity, the thumb points toward the resulting force vector.
In this specific case, the magnitude is required, which we have calculated as \( 12 \) N.
Step 4: Final Answer:
The magnetic force acting on the given charge is \( 12 \) N, which corresponds to option (B).