Electrostatics - Negative charge in sphere

In summary, the problem involves a grounded metal sphere with a negative charge at a distance a from the center. Using the method of image charges, the electric field and force on the charge can be determined. The maximal and minimal surface charge density can also be calculated using the first uniqueness theorem. However, the magnitude of the charge on the inner and outer surfaces of the sphere is still unknown when the surface potential is changed. According to Gauss's law, the total charge on the inner surface must be the negative of the charge q, and the charge on the outer surface can be found using the electric field outside the sphere.
  • #1
jacobrhcp
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Homework Statement



a metal grounded sphere of radius R in vacuum has a negative charge inside at a distance a from the centre. a) Draw the e-field in the sphere, b) find the force on the charge, c) and determine the maximal and minimal surface charge density, d) as well as the magnitude of the charge on the in,- and outside of the sphere, when we change the surface potential from zero to V.

The Attempt at a Solution



I did a-b-c, and I don't know d yet.

a-b:

I was thinking Image charges here.

if I put a charge [tex] q' = q \frac{b}{R} [/tex] at distance [tex] b=\frac{R^{2}}{a} [/tex] from the centre in a straight line from the centre and charge q, the potential is zero on the surface, just like in the problem. the first uniqueness theorem now tells me that the potential inside the sphere is uniquely determined.

The second uniquess theorem tells me the E-field is uniquely determined, because the total charge on the sphere is given (=0).

The force is now just a matter of putting symbols in formulas.

c: the surface charge is minimal/maximal on the places of the sphere closest and furthest from the image charge. So [tex]\nabla E = \frac{\rho}{e_0} [/tex] and you compute [tex]\rho[/tex]

But what is the total charge when the potential is changed?
 
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  • #2
"because the total charge on the sphere is given (=0)."

When the sphere is grounded, the total surface charge on the inner surface of the spherical shell must be the negative of the charge q. This is because V=0, means E=0 just outside. By Gauss's law the net charge must be zero.
If the sphere's surface is at V, E just outside equals V/R.
Gauss's law will then give the charge on the outer surface of the sphere.
 

Related to Electrostatics - Negative charge in sphere

What is electrostatics?

Electrostatics is a branch of physics that deals with the study of electric charges at rest.

What is a negative charge?

A negative charge is an excess of electrons in an object, resulting in an overall negative charge.

How is negative charge distributed in a sphere?

Negative charge in a sphere is distributed evenly throughout the surface of the sphere, with a higher concentration of charge at the edges and a lower concentration at the center.

What is the relationship between negative charge and electric force?

Negative charge creates an electric field around an object, which can exert a force on other charged objects in its vicinity. Like charges repel each other, so negative charge in a sphere will repel other negative charges.

How can negative charge be induced in a sphere?

Negative charge can be induced in a sphere by bringing a negatively charged object close to it, causing the electrons in the sphere to redistribute and create an overall negative charge.

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