You have landed a summber job working with an Astrophysics group investigating t

Question

You have landed a summber job working with an Astrophysics group investigating the origin of high-energy particles in the galaxy.  The grou you are joining has just discovered a large spherical nebula with a radius 1.2 million km.  The nebula consists of about 5 x 10^10 hydrgogen nuclei (protons) which appear to be uniformly distributed in the shape of a sphere.  At the center of this sphere of positive charge is a very small neutron star.  Your group had detected electrons emergin from the nebula.  A friend of yours has a theory that the electrons are coming from the neutron star.  To test that theory, she asks you to calculate the minimum speed that an electron would need to start from the neutron star and just make it to outside the nebula.  From the inside cover of your trusty physics text you find that the charge of a proton (and an electron) is 1.6 x10^-19 C, the mass of the proton is 1.7 x 10^-27 kg, and the mass of the electron is 9.1 x 10^-312 kg.

Explanation / Answer

V = k Q / R    where Q is the charge of the nebula and R the radius

By Gauss Law this is the electric potential at the edge of the nebula

E = V q = k Q q / R   where E is the energy to remove an electron

1/2 m v^2 = k Q q / R     KE required for an electron to escape

v = (2 k Q q / (R * m))^1/2

v = (2 * 9 * 10E9 * 5 * 10E10 * 1.6 * 10E-19 * 1.6 * 10E-19) / (1.2 * 10E9 * 9.1 * 10E-31))^1/2

v = 145 m/s

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