An electron (with charge \(-e\) and mass \(m_{\mathrm{e}}\) ) moving in the positive \(x\) -direction enters a velocity selector. The velocity selector consists of crossed electric and magnetic fields: \(\vec{E}\) is directed in the positive \(y\) -direction, and \(\vec{B}\) is directed in the positive \(z\) -direction. For a velocity \(v\) (in the positive \(x\) -direction \(),\) the net force on the electron is zero, and the electron moves straight through the velocity selector. With what velocity will a proton (with charge \(+e\) and mass \(m_{\mathrm{p}}=1836 \mathrm{~m}_{\mathrm{e}}\) ) move straight through the velocity selector? a) \(v\) b) \(-v\) c) \(v / 1836\) d) \(-v / 1836\)

When studying how particles move through electric and magnetic fields, the electric force plays a critical role. It's one of the fundamental forces that influence charged particles, such as electrons and protons. The strength of the electric force acting on a particle is determined by the equation
\(F_e = q \times E\),
where q represents the charge of the particle and E the electric field intensity. The direction of the force follows the direction of the electric field for a positive charge and is opposite for a negative charge. In a velocity selector, the electric force ensures that only particles with a specific velocity can pass through without deflection, as other velocities would result in the particle being pushed upward or downward by this force, deviating from the straight path.

The magnetic force is another component that acts perpendicularly to both the magnetic field and the velocity of a charged particle. This force is fundamentally described by the equation
\(F_m = q \times v \times B\),
where v is the velocity of the particle, B is the magnetic field strength, and q is again the charge. Charged particles moving in a magnetic field experience a force that directs them in a circular or helical path, according to the left-hand rule for negatively charged particles like electrons, and the right-hand rule for positively charged particles like protons. In a velocity selector, the magnetic force must balance exactly with the electric force for a particle to continue moving in a straight line without being deflected to the side.

The charge and mass of particles are two intrinsic properties that greatly influence their motion in electric and magnetic fields. The charge determines the magnitude and direction of the electric and magnetic forces acting on a particle. For instance, a particle with a negative charge like an electron will experience forces in opposite directions when compared to a positively charged particle like a proton when they move through the same electric field.

The mass of a particle, on the other hand, affects its acceleration as a result of these forces according to Newton's second law: \(F = m \times a\). Particles with larger mass will accelerate less under the same force. This is particularly important for velocity selectors, as heavier particles (like protons) need different settings compared to lighter particles (like electrons) to move straight through due to their different mass-to-charge ratios. Thus, while an electron can pass through with velocity \(v\), a proton, which has 1836 times the mass of an electron, would theoretically pass through with a velocity \(v/1836\) if the selection was purely based on mass and charge without considering the existing forces.