Electric field strength `vecE = E_0 hati ` and `vecB = B_0 hati` exists in a region. A charge is projected with a velocity `vecv = v_0 hatj ` origin, then

Options

A. It moves along helix with constant pitch
B. It moves along circular path in Y Z plane
C. It moves along helix with increasing pitch
D. It moves along helix with decreasing pitch

Explanation

To solve the problem, we need to analyze the motion of a charged particle projected in the presence of both an electric field and a magnetic field. Here's a step-by-step solution:

### Step 1: Understand the Forces Acting on the Charge
The charged particle experiences two forces:
1. **Electric Force (\( \vec{F_E} \))**: This force is given by \( \vec{F_E} = q \vec{E} \), where \( q \) is the charge and \( \vec{E} = E_0 \hat{i} \) is the electric field.
2. **Magnetic Force (\( \vec{F_B} \))**: This force is given by \( \vec{F_B} = q (\vec{v} \times \vec{B}) \), where \( \vec{B} = B_0 \hat{i} \) is the magnetic field and \( \vec{v} = v_0 \hat{j} \) is the velocity of the charge.

### Step 2: Calculate the Electric Force
The electric force acting on the charge is:
\[ \vec{F_E} = q E_0 \hat{i} \]
This force acts in the direction of the electric field, which is along the \( \hat{i} \) direction.

### Step 3: Calculate the Magnetic Force
The magnetic force can be calculated using the cross product:
\[ \vec{F_B} = q (\vec{v} \times \vec{B}) = q (v_0 \hat{j} \times B_0 \hat{i}) \]
Using the right-hand rule, we find:
\[ \hat{j} \times \hat{i} = -\hat{k} \quad \Rightarrow \quad \vec{F_B} = -q v_0 B_0 \hat{k} \]
This force acts in the negative \( \hat{k} \) direction.

### Step 4: Determine the Resultant Force
The total force acting on the charge is the vector sum of the electric and magnetic forces:
\[ \vec{F} = \vec{F_E} + \vec{F_B} = q E_0 \hat{i} - q v_0 B_0 \hat{k} \]

### Step 5: Analyze the Motion
The charge will experience:
- An acceleration in the \( \hat{i} \) direction due to the electric force.
- A circular motion in the \( \hat{j} \) and \( \hat{k} \) plane due to the magnetic force.

### Step 6: Describe the Path of the Charge
The charge will move in a helical path because:
- The electric field causes a constant acceleration in the \( \hat{i} \) direction, which increases the velocity in that direction.
- The magnetic field causes circular motion in the \( \hat{j} \) and \( \hat{k} \) plane.

Since the velocity in the \( \hat{i} \) direction is increasing due to the electric field, the pitch of the helix (the distance between successive turns) will also increase.

### Conclusion
The correct description of the motion of the charge is that it moves along a helix with increasing pitch.

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