91Ó°ÊÓ

The velocity of electrons in electric and magnetic fields is vital to understanding the behavior of charged particles under simultaneous field influences. When an electron moves through an electric field (\( E \)) and a magnetic field (\( B \)), it experiences forces due to each field. The crucial condition where there's no deflection occurs when these forces exactly cancel each other out. This is expressed mathematically where the force from the electric field, \( F_E = eE \), equals the force from the magnetic field, \( F_B = evB \), noting \( e \) is the electron charge and \( v \) is the electron velocity. In the case without deflection, the relationship simplifies to \[ v = \frac{E}{B} \]. Plugging in the given values: \( E = 1.56 \times 10^{4} \, \text{V/m} \) and \( B = 4.62 \times 10^{-3} \, \text{T} \), the velocity of electrons in this specific setup is found to be approximately \( 3.38 \times 10^{6} \, \text{m/s} \). Key points include:

• Electron velocity is determined by balancing electric and magnetic field forces.
• Speed remains constant as path deflection is neutralized.
• Velocity calculation becomes a simple operation involving division of field magnitudes.

The magnetic force on a moving electron in a magnetic field acts perpendicularly to both the velocity of the electron and the magnetic field lines. This is a fundamental principle known as the Lorentz force law. The force can be expressed as: \[ F_B = evB \sin\theta \], where \( \theta \) is the angle between the velocity vector \( \overrightarrow{v} \) and the magnetic field vector \( \overrightarrow{B} \). In situations where the electron's path is perpendicular to the magnetic field, \( \theta = 90^\circ \) and \( \sin\theta = 1 \), simplifying our formula.When no deflection is observed, this magnetic force is perfectly balanced by the electric force. If the electric field is removed but the magnetic field is maintained, the electrons instead follow a circular path due to the magnetic force. The expression for the radius \( r \) of this path is derived from balancing centripetal force with the magnetic force:\[ evB = \frac{mv^2}{r} \]which rearranges to:\[ r = \frac{mv}{eB} \]. This shows how:

• Magnetic forces redirect electron motion under field influence, causing circular or spiral paths.
• Force magnitude can determine path curvature or deviation.

Electric fields are regions around a charged particle where forces are exerted on other charges. In the context of electron motion, the electric field exerts a force that can alter the trajectory of the electron. The force from an electric field is given by: \[ F_E = eE \], where \( e \) is the magnitude of the electron charge and \( E \) is the electric field strength.Understanding the interaction between an electric field and electrons provides insights into how they can be manipulated or controlled. For instance, in devices where precise control over electron paths is required, such as cathode ray tubes or electron microscopes, the electric field plays a pivotal role.Key insights regarding electric fields include:

• They apply a constant force on charges, influencing acceleration and direction.
• Magnitude and direction of electric fields determine how and where electrons move within a field.