Question
How does an electron move in a straight line between charged plates with both electric and magnetic fields present?
Answers
Answered by
GPT-4o mini
To understand how an electron moves in a straight line between charged plates while subject to both electric and magnetic fields, we need to consider the forces that act on the electron due to these fields.
1. **Electric Field**: When there are charged plates, an electric field \(\vec{E}\) is established between them. The direction of the electric field is from the positively charged plate to the negatively charged plate. An electron, which has a negative charge, will experience a force due to the electric field given by:
\[
\vec{F}_E = q\vec{E}
\]
where \(q\) is the charge of the electron (\(q = -e\), where \(e\) is the elementary charge). The force acting on the electron due to the electric field will thus be directed opposite to the direction of the electric field.
2. **Magnetic Field**: If a magnetic field \(\vec{B}\) is also present, it exerts a magnetic force on the moving electron. The magnetic force \(\vec{F}_B\) can be calculated using the Lorentz force law:
\[
\vec{F}_B = q(\vec{v} \times \vec{B})
\]
where \(\vec{v}\) is the velocity of the electron. The direction of the magnetic force is given by the right-hand rule, and it will be perpendicular to both the velocity of the electron and the magnetic field.
3. **Equilibrium Condition for Straight-Line Motion**: For the electron to move in a straight line, the resultant forces acting on it must balance such that the net force is zero:
\[
\vec{F}_E + \vec{F}_B = 0.
\]
This means that the electric force must equal in magnitude and opposite in direction to the magnetic force:
\[
q\vec{E} + q(\vec{v} \times \vec{B}) = 0.
\]
After simplification, this condition can be expressed as:
\[
\vec{E} + (\vec{v} \times \vec{B}) = 0.
\]
Rearranging gives us:
\[
\vec{E} = -(\vec{v} \times \vec{B}).
\]
4. **Velocity and Fields**: The above condition indicates that the velocity of the electron must be such that the magnetic force counteracts the electric force. The speed of the electron (\(v\)) will determine the magnetic force for a given magnetic field. The angle between \(\vec{v}\) and \(\vec{B}\) will also affect the magnetic force calculations.
5. **Result**: If the electron is initially moving at the right speed and direction, it can achieve a state where it moves in a straight line between the plates. The straight-line path is a result of the electric and magnetic forces canceling each other out, allowing the electron to continue along that trajectory without accelerating.
In summary, for an electron to maintain straight-line motion between charged plates in the presence of both electric and magnetic fields, the forces from these fields must balance in such a way that the net force on the electron is zero. The relationship between the electric field, magnetic field, and the velocity of the electron will dictate the specific conditions under which this balance occurs.
1. **Electric Field**: When there are charged plates, an electric field \(\vec{E}\) is established between them. The direction of the electric field is from the positively charged plate to the negatively charged plate. An electron, which has a negative charge, will experience a force due to the electric field given by:
\[
\vec{F}_E = q\vec{E}
\]
where \(q\) is the charge of the electron (\(q = -e\), where \(e\) is the elementary charge). The force acting on the electron due to the electric field will thus be directed opposite to the direction of the electric field.
2. **Magnetic Field**: If a magnetic field \(\vec{B}\) is also present, it exerts a magnetic force on the moving electron. The magnetic force \(\vec{F}_B\) can be calculated using the Lorentz force law:
\[
\vec{F}_B = q(\vec{v} \times \vec{B})
\]
where \(\vec{v}\) is the velocity of the electron. The direction of the magnetic force is given by the right-hand rule, and it will be perpendicular to both the velocity of the electron and the magnetic field.
3. **Equilibrium Condition for Straight-Line Motion**: For the electron to move in a straight line, the resultant forces acting on it must balance such that the net force is zero:
\[
\vec{F}_E + \vec{F}_B = 0.
\]
This means that the electric force must equal in magnitude and opposite in direction to the magnetic force:
\[
q\vec{E} + q(\vec{v} \times \vec{B}) = 0.
\]
After simplification, this condition can be expressed as:
\[
\vec{E} + (\vec{v} \times \vec{B}) = 0.
\]
Rearranging gives us:
\[
\vec{E} = -(\vec{v} \times \vec{B}).
\]
4. **Velocity and Fields**: The above condition indicates that the velocity of the electron must be such that the magnetic force counteracts the electric force. The speed of the electron (\(v\)) will determine the magnetic force for a given magnetic field. The angle between \(\vec{v}\) and \(\vec{B}\) will also affect the magnetic force calculations.
5. **Result**: If the electron is initially moving at the right speed and direction, it can achieve a state where it moves in a straight line between the plates. The straight-line path is a result of the electric and magnetic forces canceling each other out, allowing the electron to continue along that trajectory without accelerating.
In summary, for an electron to maintain straight-line motion between charged plates in the presence of both electric and magnetic fields, the forces from these fields must balance in such a way that the net force on the electron is zero. The relationship between the electric field, magnetic field, and the velocity of the electron will dictate the specific conditions under which this balance occurs.