As a charged particle 'q' moving with a velocity `vec(v)` enters a uniform magnetic field `vec(B)`, it experience a force `vec(F) = q(vec(v) xx vec(B)). For theta = 0^(@) or 180^(@), theta` being the angle between `vec(v) and vec(B)`, force experienced is zero and the particle passes undeflected. For `theta = 90^(@)`, the particle moves along a circular arc and the magnetic force (qvB) provides the necessary centripetal force `(mv^(2)//r)`. For other values of `theta (theta !=0^(@), 180^(@), 90^(@))`, the charged particle moves along a helical path which is the resultant motion of simultaneous circular and translational motions.
Suppose a particle that carries a charge of magnitude q and has a mass `4 xx 10^(-15)` kg is moving in a region containing a uniform magnetic field `vec(B) = -0.4 hat(k) T`. At some instant, velocity of the particle is `vec(v) = (8 hat(i) - 6 hat(j) 4 hat(k)) xx 10^(6) m s^(-1)` and force acting on it has a magnitude 1.6 N
Motion of charged particle will be along a helical path with

Options

A. A translational component along x-direction and a circular component in the y-z plane
B. A translational component along y-direction and a circular component in the x-z plane
C. A translational component along z-axis and a circular component in the x-y plane
D. Direction of translational component and plane of circular component are uncertain

Explanation

To solve the problem of a charged particle moving in a uniform magnetic field, we will break down the solution step by step.

### Step 1: Identify the Given Information
We have the following information:
- Charge of the particle, \( q \) (not specified in the problem).
- Mass of the particle, \( m = 4 \times 10^{-15} \, \text{kg} \).
- Magnetic field, \( \vec{B} = -0.4 \hat{k} \, \text{T} \).
- Velocity of the particle, \( \vec{v} = (8 \hat{i} - 6 \hat{j} + 4 \hat{k}) \times 10^6 \, \text{m/s} \).
- Magnitude of the force acting on the particle, \( F = 1.6 \, \text{N} \).

### Step 2: Calculate the Components of Velocity
The velocity vector can be broken down into its components:
- \( v_x = 8 \times 10^6 \, \text{m/s} \) (in the x-direction),
- \( v_y = -6 \times 10^6 \, \text{m/s} \) (in the y-direction),
- \( v_z = 4 \times 10^6 \, \text{m/s} \) (in the z-direction).

### Step 3: Determine the Angle Between Velocity and Magnetic Field
The magnetic field is directed along the z-axis. The angle \( \theta \) between the velocity vector and the magnetic field can be determined by considering the components of the velocity:
- The component of velocity parallel to the magnetic field is \( v_z \).
- The components perpendicular to the magnetic field are \( v_x \) and \( v_y \).

### Step 4: Analyze the Motion of the Charged Particle
Since the particle has components of velocity both parallel and perpendicular to the magnetic field:
- The component \( v_z \) (parallel to \( \vec{B} \)) will cause translational motion along the z-axis.
- The components \( v_x \) and \( v_y \) (perpendicular to \( \vec{B} \)) will cause circular motion in the xy-plane.

### Step 5: Conclusion on the Motion Path
Thus, the motion of the charged particle will be a combination of:
- Circular motion in the xy-plane due to the perpendicular components of velocity,
- Translational motion along the z-axis due to the parallel component of velocity.

### Final Answer
The motion of the charged particle will be along a helical path, with circular motion in the xy-plane and translational motion along the z-axis.

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