Electric field and magnetic field in a region of space are given by `vec E = E_(0) hat I " and " vec B = - B_(0) hat k `, respectively. A positively charged particle `(+q)` is released from rest at the origin. When the particle reaches a point `P(x, y, z)` , then it attains kinetic energy K that is equal to

Options

A. `qB_(0) x`
B. `qB_(0)y`
C. `qE_(0)y`
D. `qE_(0)x`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the Forces Acting on the Particle
The positively charged particle \( +q \) is subjected to an electric field \( \vec{E} = E_0 \hat{i} \) and a magnetic field \( \vec{B} = -B_0 \hat{k} \). Since the particle is released from rest, its initial velocity \( \vec{v} = 0 \).

**Hint:** Remember that the magnetic force on a charged particle is given by \( \vec{F}_B = q \vec{v} \times \vec{B} \). If the particle is at rest, this force will be zero.

### Step 2: Calculate the Electric Force
The electric force \( \vec{F}_E \) acting on the particle due to the electric field is given by:
\[ \vec{F}_E = q \vec{E} = q E_0 \hat{i} \]

**Hint:** The electric force is in the direction of the electric field.

### Step 3: Apply the Work-Energy Theorem
According to the work-energy theorem, the work done by all forces on the particle is equal to the change in kinetic energy. Since the magnetic force does no work (it is always perpendicular to the displacement), we only need to consider the work done by the electric force.

The work done \( W \) by the electric force as the particle moves from the origin to a point \( P(x, y, z) \) is given by:
\[ W = \vec{F}_E \cdot \vec{d} = F_E \cdot d_x = (q E_0) \cdot x \]
where \( d_x \) is the displacement along the x-direction.

**Hint:** The work done by a force is the product of the force and the displacement in the direction of the force.

### Step 4: Relate Work Done to Kinetic Energy
Since the particle starts from rest, its initial kinetic energy is zero. Therefore, the change in kinetic energy \( K \) is equal to the work done:
\[ K = W = q E_0 x \]

**Hint:** Remember that the initial kinetic energy is zero because the particle starts from rest.

### Final Result
Thus, the kinetic energy \( K \) of the particle when it reaches point \( P(x, y, z) \) is:
\[ K = q E_0 x \]

### Summary
The final kinetic energy of the charged particle when it reaches point \( P(x, y, z) \) is given by:
\[ K = q E_0 x \]