In which condition an electron will move undeflected in the presence of crossed electric and magnetic field ?

Options

A. When electric and magnetic field in parallel
B. When Electric and magnetic field in perpendicular
C. Both (a) and (b)
D. None of them

Answer

To determine the condition under which an electron moves undeflected in the presence of crossed electric and magnetic fields, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Forces on a Charged Particle**:
When a charged particle, such as an electron, moves through an electric field (E) and a magnetic field (B), it experiences two forces: the electric force (F_E) and the magnetic force (F_B).

2. **Electric Force Calculation**:
The electric force acting on the electron is given by:
\[ F_E = qE \]
where \( q \) is the charge of the electron (which is negative) and \( E \) is the electric field strength.

3. **Magnetic Force Calculation**:
The magnetic force acting on the electron is given by the Lorentz force equation:
\[ F_B = q(\mathbf{v} \times \mathbf{B}) \]
where \( \mathbf{v} \) is the velocity of the electron and \( \mathbf{B} \) is the magnetic field.

4. **Condition for No Deflection**:
For the electron to move undeflected, the net force acting on it must be zero. This means that the electric force must be equal in magnitude and opposite in direction to the magnetic force:
\[ F_E + F_B = 0 \quad \Rightarrow \quad F_E = -F_B \]

5. **Substituting Forces**:
Substituting the expressions for the forces gives:
\[ qE = -q(\mathbf{v} \times \mathbf{B}) \]
Since \( q \) is negative (for an electron), we can simplify this to:
\[ E = \mathbf{v} \times \mathbf{B} \]

6. **Direction of Fields**:
The electric field (E) and the magnetic field (B) must be perpendicular to each other for the cross product \( \mathbf{v} \times \mathbf{B} \) to yield a force that can balance the electric force. Thus, the condition for the electron to be undeflected is that the electric field and magnetic field are perpendicular to each other.

7. **Conclusion**:
Therefore, the condition under which an electron will move undeflected in the presence of crossed electric and magnetic fields is when the electric field and magnetic field are perpendicular to each other.

### Final Answer:
The electron will move undeflected when the electric field and magnetic field are perpendicular to each other.

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