4. An electron and a proton move in a uniform magnetic field with same speed directed perpendicular to the magnetic field. They experience forces Fe and Fp respectively in mutually opposite directions, where Fp = 1840 Fe 1 point True False
Answers
Answer:
The reason that the particles would be deflected into circular paths is because of the magnetic force induced by the magnetic field is perpendicular to both the particles’ velocities and the magnetic field. The magnitude of this force can be found by the equation
F=qvBsin(θ)F=qvBsin(θ)
where F = the magnetic force,
q = the charge of the particles,
v = the velocity,
B = the magnetic field strength in which the particles move,
and θ = the angle between the velocity of the charge and the magnetic field.
For this problem, we’re going to assume θ=π2θ=π2 radians or 90° because the problem doesn’t say otherwise, making our new magnitude equation:
F=qvBF=qvB
Since the particles have the same magnitude of charge, velocity, and magnetic field strengths acting on them, the forces acting on either particle will be the same in magnitude, just in opposite directions because they are opposite charges.
Note/Short Digression: The reason the forces act in opposite directions is because the equation for the magnitudes is derived from an equation for the vectors that contains a cross product:
F⃗ =qv⃗ ×B⃗ F→=qv→×B→
Because of this cross product, the force on either particle will always be perpendicular to its motion, making the force on the particle centripetal and the path each particle follows a circle. Because the proton and electron have opposite charges, the vector of qv⃗ qv→ has a direction for each before going into the cross product, causing the force to act in the opposite directions for each. The proton follows the right-hand rule, and the electron goes in the opposite direction (which can also be found by a “left-hand rule, if you were to make one). - End of Note/Short Digression
Despite the opposite directions, as aforementioned, the particles have the same magnitude, which is the only important factor in determining the radii of the particle’s circular paths. The difference in the direction of the force would only make a difference in clockwise/counter-clockwise motion. Despite the fact that the same force acts on each particle, this does not mean the particles have the amount of resultant motion.
Because F=maF=ma, a more massive object doesn’t accelerate as much as a less massive object does from a force of a equal magnitude acting on either object (unless you get into relativity, where there is a special case where this isn’t true). Since protons are more massive than electrons, the electron will be accelerated by the magnetic force more than the proton. Stronger centripetal acceleration = smaller circular path, so the radius of the electron’s path will be smaller than that of the proton’s path.
I hope that helps!