Minimal Launch Velocity needed

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SUMMARY

The minimal launch velocity required for a negatively charged particle to escape from a positively charged dielectric sphere is determined by the potential energy at the surface of the sphere and the energy needed to reach infinity. The potential energy at the surface can be calculated using the charge density (ρ) and the radius (R) of the sphere. If the radius R is doubled, the minimal launch velocity increases due to the increased potential energy associated with the larger sphere.

PREREQUISITES
  • Understanding of electrostatics and electric potential
  • Familiarity with the concept of escape velocity
  • Knowledge of potential energy calculations in electric fields
  • Basic physics of charged particles and forces
NEXT STEPS
  • Calculate the potential energy of a charged particle in an electric field
  • Study the concept of escape velocity in gravitational and electric fields
  • Explore the effects of charge density on electric potential
  • Investigate the relationship between radius and potential energy in charged spheres
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Students and educators in physics, particularly those focusing on electrostatics, as well as researchers exploring particle dynamics in electric fields.

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Homework Statement



A small negatively charged particle (mass m, charge –q) is currently at rest at the surface of a large positively and uniformly charged dielectric sphere (radius R, charge density p). What minimal “Launch velocity” v0 should be given to this particle so that it could leave the attracting “mother” sphere to travel to infinitely remote places? If the radius R of the sphere is increased by a factor of 2 (everything else left the same), how would the minimal launch velocity change?

Homework Equations



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The Attempt at a Solution



Please help!
 
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Hints:

What is the potential energy of the particle currently (at distance R)?

What will the potential energy of the particle be if it really does manage to fly off to infinity?
 
Suppose that the particle is at a distance [itex]r \ge R[/itex] from the center of the sphere.
What is the force exerted on it?

Also, have you ever seen the concept of 'escape velocity' (e.g. for rockets in a gravitational field)?
 

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