Calculating Magnetic Field Needed for Proton Beam

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To keep a proton beam moving horizontally in a uniform magnetic field while counteracting gravity, the magnetic force must equal the gravitational force acting on the protons. The relevant equation is B = qv/F, where B is the magnetic field strength, q is the charge of the proton, v is its velocity, and F is the gravitational force. The discussion emphasizes the need to equate the gravitational force (Fg) with the magnetic force (Fb) to find the required magnetic field strength. Participants note that additional information may be necessary to complete the calculations. The solution involves substituting known values into the equations to derive the magnetic field strength needed.
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Homework Statement


A beam of protons is accelerated to a speed of 5.0x10^6 m/s in a particle accelerator into a uniform magnetic field. What B field perpendicular to the velocity of the proton would cancel the force of gravity and keep the beam moving exactly horizontally?

Homework Equations


B = qv/F


The Attempt at a Solution


I haven't made any attempts because I don't think that I was given enough information...
 
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Begin with Fg = Fb (force of gravity equals magnetic force).
Fill in the detailed formulas, then solve for what you are looking for.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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