A charge particle is moving along positive y-axis in uniform electric and magnetic fields
`E=E_0hatk`
and `B=B_0hati`
Here `E_0` and `B_0` are positive constants.choose the correct options.

Options

A. particle may be deflected towards positive x-axis
B. particle may be deflected towards negative z-axis
C. particle may pass undeflected
D. kinetic energy of particle may remain constant

Explanation

To analyze the motion of a charged particle moving in uniform electric and magnetic fields, we will follow these steps:

### Step 1: Identify the Given Information
The electric field \( \mathbf{E} \) is given as \( E = E_0 \hat{k} \) (along the positive z-axis), and the magnetic field \( \mathbf{B} \) is given as \( B = B_0 \hat{i} \) (along the positive x-axis). The charged particle is moving along the positive y-axis.

### Step 2: Determine the Forces Acting on the Charged Particle
1. **Electric Force**: The electric force \( \mathbf{F_E} \) acting on the particle with charge \( q \) in the electric field is given by:
\[ \mathbf{F_E} = q \mathbf{E} = q E_0 \hat{k} \]
This force acts in the positive z-direction.

2. **Magnetic Force**: The magnetic force \( \mathbf{F_B} \) acting on the particle moving with velocity \( \mathbf{v} \) in the magnetic field is given by:
\[ \mathbf{F_B} = q (\mathbf{v} \times \mathbf{B}) \]
Here, the velocity \( \mathbf{v} \) is along the positive y-axis, \( \mathbf{v} = v \hat{j} \), and the magnetic field is along the positive x-axis. Therefore, we calculate the cross product:
\[ \mathbf{F_B} = q (v \hat{j} \times B_0 \hat{i}) = q v B_0 (\hat{j} \times \hat{i}) = -q v B_0 \hat{k} \]
The magnetic force acts in the negative z-direction.

### Step 3: Analyze the Conditions for Deflection
1. **If the electric force and magnetic force are equal**:
- For the particle to pass undeflected, the net force must be zero:
\[ \mathbf{F_E} + \mathbf{F_B} = 0 \implies q E_0 \hat{k} - q v B_0 \hat{k} = 0 \]
This implies:
\[ E_0 = v B_0 \]
In this case, the particle will not be deflected.

2. **If \( E_0 > v B_0 \)**:
- The electric force will dominate, and the particle will be deflected toward the positive z-axis.

3. **If \( E_0 < v B_0 \)**:
- The magnetic force will dominate, and the particle will be deflected toward the negative z-axis.

### Step 4: Kinetic Energy Consideration
- The kinetic energy of the particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
If the particle is undeflected, its speed remains constant, and thus its kinetic energy remains constant.

### Conclusion
Based on the analysis:
- The particle may be deflected toward the positive z-axis if the electric force is greater.
- The particle may be deflected toward the negative z-axis if the magnetic force is greater.
- The particle may pass undeflected if the forces are equal.
- The kinetic energy of the particle may remain constant if it passes undeflected.

### Correct Options
- The particle may be deflected toward the positive z-axis.
- The particle may be deflected toward the negative z-axis.
- The particle may pass undeflected.
- The kinetic energy of the particle may remain constant.