Point Charge in a Glass Sphere

[Lunar_Lander]

Homework Statement

There is a point charge Q = 4π nC inside a glass sphere which has the radius 1 m and the dielectric constant ε_r=10. This sphere is enclosed in a hollow metal sphere with an inner radius of 2 m and a thickness of 0.5 m.

The task is to find the amount of the electric field strength in the glass sphere, in the gap between glass and metal, in the metal layer and on the outside.

Homework Equations

Gauss law of electrostatics.

The Attempt at a Solution

As far as I know, the field outside of the metal sphere should be as if the sphere would be a point charge, i.e.E(r)=Q/(4πε₀r²). But I am a bit lost at the other parts. Is the field in the glass sphere for instance simply: E(r)=Q/(4π ε_rε₀r²)?

 

[collinsmark]

Homework Statement

There is a point charge Q = 4π nC inside a glass sphere which has the radius 1 m and the dielectric constant ε_r=10. This sphere is enclosed in a hollow metal sphere with an inner radius of 2 m and a thickness of 0.5 m.

The task is to find the amount of the electric field strength in the glass sphere, in the gap between glass and metal, in the metal layer and on the outside.

Homework Equations

Gauss law of electrostatics.

The Attempt at a Solution

As far as I know, the field outside of the metal sphere should be as if the sphere would be a point charge, i.e.E(r)=Q/(4πε₀r²).

Correct. (Of course this is only correct due to the spherical symmetry of the whole thing. If it were any other shape, the answer would be different.)

But I am a bit lost at the other parts. Is the field in the glass sphere for instance simply: E(r)=Q/(4π ε_rε₀r²)?

Yes, the equation for inside the glass sphere is also correct.

You should be able to use the standard version of Gauss' law for the region (gap) between the glass sphere and hollow metal shell (ε_r = 1 in this region).

The electric field within a conductor is special [Edit: by that I mean within the conducting material itself]. But I'll leave that for you to look up.

 

[Lunar_Lander]

OK, thanks :)! In the metal, there should be a shift of charges due to the field from the inside, so that the inside of the metal layer is free of the electric field, i.e. |E(r)|=0?

One last pointer on the fields inside though, please. Do I need to incoporate the radius of the glass sphere into the calculation somehow? Or can I just say that below r₀ (which is the radius of the glass sphere, the inner radius of the metal was denoted as r₁ and the thickness was given as d, so the outer surface of the metal sphere is at r₁+d), the field is E(r<r₀)=Q/(4π*ε₀*ε_r*r²), then in the air gap, the field is E(r₀<r<r₁)=Q/(4π*ε₀*r²)?

And accordingly, E(r>r₁+d)=Q/(4π*ε₀*r²)?

 

[collinsmark]

OK, thanks :)! In the metal, there should be a shift of charges due to the field from the inside, so that the inside of the metal layer is free of the electric field, i.e. |E(r)|=0?

Yes that's right. The (static) electric field within a conductor is always zero. Things get more complicated when you move on to electrodynamics. But with static charges, the field within a conducting material is always zero.

One last pointer on the fields inside though, please. Do I need to incoporate the radius of the glass sphere into the calculation somehow? Or can I just say that below r₀ (which is the radius of the glass sphere, the inner radius of the metal was denoted as r₁ and the thickness was given as d, so the outer surface of the metal sphere is at r₁+d), the field is E(r<r₀)=Q/(4π*ε₀*ε_r*r²),

The latter looks good to me.

then in the air gap, the field is E(r₀<r<r₁)=Q/(4π*ε₀*r²)?

Yes, that's right! And that's for the same reason that you can say E = Q/(4π*ε₀*r²) outside of the metal shell too. As long as there is perfect spherical symmetry, it doesn't matter what is on the outside of the Gaussian surface. All that matters is the total charge within.

And accordingly, E(r>r₁+d)=Q/(4π*ε₀*r²)?

Yes. Good work.

 

[rwisz]

Thank you for your question. I would approach this problem by first understanding the basic principles involved. In this case, we are dealing with electrostatics, which is the study of electric charges at rest. The key equation we will use is Gauss's law, which relates the electric flux through a closed surface to the enclosed charge.

In this problem, we have a point charge Q located inside a glass sphere with a dielectric constant εr = 10. The glass sphere is then enclosed in a hollow metal sphere with an inner radius of 2 m and a thickness of 0.5 m. Our goal is to find the electric field strength at various points in and around the system.

To begin, we can use Gauss's law to find the electric field strength inside the glass sphere. Since the glass sphere is a closed surface, the electric flux through it is equal to the enclosed charge divided by the permittivity of free space (ε0). Therefore, we can write:

∮E⃗⋅dA⃗ = Q/ε0

Where ∮E⃗⋅dA⃗ is the electric flux through the surface of the glass sphere. Since the electric field is constant and perpendicular to the surface of the sphere, we can simplify this equation to:

E(r)4πr2 = Q/ε0

Where E(r) is the electric field strength at a distance r from the point charge. Since we know the value of Q and the radius of the glass sphere, we can solve for E(r):

E(r) = Q/(4πε0r2)

This is the electric field strength inside the glass sphere. Notice how the dielectric constant εr does not appear in this equation. This is because the electric field inside a dielectric material is reduced by a factor of εr compared to the electric field in free space. Therefore, the electric field inside the glass sphere is simply the electric field due to the point charge Q, divided by the dielectric constant εr.

Now, to find the electric field in the gap between the glass and metal spheres, we can use a similar approach. The electric flux through the surface of the gap is equal to the charge enclosed by the glass sphere and the metal sphere, divided by ε0. Therefore, we can write:

∮E⃗⋅dA⃗ = (Q+Q