8 An electric field of \( 1.50 \mathrm{kV} / \mathrm{m} \) and a perpendicular magnetic field of 0.400 T act on a moving electron to produce no net force. What is the electron's speed? no net force means electric force and magnetic field are equal to each other \[ F_{e}=F_{B} \] \[ q E=q v B \]

To determine the speed of the electron when the electric force equals the magnetic force, we start with the equation \( F_{e} = F_{B} \). Given that \( F_{e} = qE \) and \( F_{B} = qvB \), we can set these equal to each other: \[ qE = qvB \] Since \( q \) (the charge of the electron) appears on both sides, we can cancel it out (assuming it’s non-zero) and rearrange to find the electron's speed (\( v \)): \[ E = vB \] Now, solving for \( v \) gives us: \[ v = \frac{E}{B} \] Substituting the values for the electric field (\( E = 1.50 \, \text{kV/m} = 1500 \, \text{V/m} \)) and the magnetic field (\( B = 0.400 \, T \)): \[ v = \frac{1500 \, \text{V/m}}{0.400 \, T} = 3750 \, \text{m/s} \] So, the speed of the electron is \( 3750 \, \text{m/s} \). Isn’t it wild to think that an electron just zipping along at \( 3750 \, \text{m/s} \) can dance perfectly in synchrony with the forces around it? Whether in circuits or plasma physics, particles like these are the unsung heroes of our modern technologies! For anyone curious about the electric and magnetic forces, these concepts give rise to phenomena like the Hall effect, where the voltage is generated across a conductor when it's subjected to a magnetic field. This principle is intertwined with countless applications, from sensors to medical imaging devices. It’s like a magic trick of physics that has turned into tangible technology!