An electron is moving along positive x-axis `A` unifrom electric field exists towards negative y-axis. What should be the direction of magnetic field of suitable magnitude so that net force of electrons is zero .

Options

A. positive z- axis
B. negative z-axis
C. positive y-axis
D. negative y-axis

Answer

To solve the problem, we need to analyze the forces acting on the electron and determine the direction of the magnetic field that will result in a net force of zero. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces Acting on the Electron
The electron is subject to two forces:
1. **Electric Force (F_E)**: This force is due to the electric field (E) and is given by the equation:
\[ F_E = qE \]
where \( q \) is the charge of the electron (which is negative).

2. **Magnetic Force (F_B)**: This force is due to the magnetic field (B) and is given by the equation:
\[ F_B = q(\mathbf{v} \times \mathbf{B}) \]
where \( \mathbf{v} \) is the velocity of the electron.

### Step 2: Determine the Direction of the Electric Field
The problem states that there is a uniform electric field directed towards the negative y-axis. Therefore, we can represent the electric field as:
\[ \mathbf{E} = -E \hat{j} \]
This means the electric force on the electron (which has a negative charge) will be directed in the positive y-direction:
\[ F_E = qE = -e(-E \hat{j}) = eE \hat{j} \]

### Step 3: Determine the Direction of the Velocity
The electron is moving along the positive x-axis, so we can represent its velocity as:
\[ \mathbf{v} = v \hat{i} \]

### Step 4: Set Up the Condition for Zero Net Force
For the net force on the electron to be zero, the sum of the electric force and the magnetic force must equal zero:
\[ F_E + F_B = 0 \]
This implies:
\[ F_B = -F_E \]

### Step 5: Calculate the Magnetic Force
The magnetic force can be expressed as:
\[ F_B = q(\mathbf{v} \times \mathbf{B}) = -e(\mathbf{v} \times \mathbf{B}) \]
Substituting the expressions for \( \mathbf{v} \) and \( \mathbf{B} \):
\[ F_B = -e(v \hat{i} \times \mathbf{B}) \]

### Step 6: Determine the Direction of the Magnetic Field
To find the direction of \( \mathbf{B} \), we need \( \mathbf{v} \times \mathbf{B} \) to point in the direction of \( -F_E \), which is in the positive y-direction (\( \hat{j} \)).

Using the right-hand rule for the cross product:
- Point your fingers in the direction of \( \mathbf{v} \) (positive x-axis).
- Curl your fingers towards the direction of \( \hat{j} \) (positive y-axis).

This means your thumb (which represents the direction of \( \mathbf{B} \)) must point in the negative z-direction (\( -\hat{k} \)). Therefore, we conclude:
\[ \mathbf{B} = -B \hat{k} \]

### Step 7: Conclusion
The direction of the magnetic field should be in the negative z-direction (or \( -\hat{k} \)) to ensure that the net force on the electron is zero.