Charged sphere

[jnuz73hbn]

Homework Statement

Hello guys,

Consider a spherical, electrically charged body with mass
and charge
, located next to a grounded conducting surface.
The interaction is modeled using the method of image charges.

(1)Determine the maximum radius for which the body is still held against
the surface by electrostatic attraction.

The charged spherical body is now freely suspended in air.
(2) Find out the critical electric field strength of air.
(3)Determine the minimum radius for which the electric field at the
surface of the body does not exceed

Relevant Equations

$$
F_{\mathrm{el}}=F_g
$$

$$
r_{\max}=\sqrt{\frac{kq^2}{4mg}}
$$

$$
E_{\mathrm{crit}}=3.0\times10^6\,\mathrm{V/m}
$$

$$
E(r_f\mid r_{\min})=E_{\mathrm{crit}}
$$





I think I'm missing something in the formulas

$$
r_{\max}=\sqrt{\frac{kq^2}{4mg}}
$$

$$
r_{\min}=\sqrt{\frac{k|q|}{E_{\mathrm{crit}}}}
$$

 

[kuruman]

Why do you think you are missing something in the formulas?

 

[jnuz73hbn]

Why do you think you are missing something in the formulas?

for the first I calculated this. Is that formula correct for r_max? Because 0,7m for a radius of a sphere is very big.
$$
r_{\max}
=
\sqrt{
\frac{
(8.99\times10^9\,\mathrm{N\,m^2/C^2})
(3.9\times10^{-7}\,\mathrm{C})^2
}{
4(2.8\times10^{-3}\,\mathrm{kg})(9.81\,\mathrm{m/s^2})
}
}
\approx 0.70\,\mathrm{m}
$$

 

[kuruman]

for the first I calculated this. Is that formula correct for r_max? Because 0,7m for a radius of a sphere is very big.
$$
r_{\max}
=
\sqrt{
\frac{
(8.99\times10^9\,\mathrm{N\,m^2/C^2})
(3.9\times10^{-7}\,\mathrm{C})^2
}{
4(2.8\times10^{-3}\,\mathrm{kg})(9.81\,\mathrm{m/s^2})
}
}
\approx 0.70\,\mathrm{m}
$$

Where did the numbers come from? You should post the problem exactly as was given to you. Also, these numbers that you show do not produce the answer that you quote. Redo the calculation, preferably on a spreadsheet, where you can easily troubleshoot what you are doing.

 

[jnuz73hbn]

Where did the numbers come from? You should post the problem exactly as was given to you. Also, these numbers that you show do not produce the answer that you quote. Redo the calculation, preferably on a spreadsheet, where you can easily troubleshoot what you are doing.

Given values are: q= 3.9 * 10^-7 C , m= 2,8 *10^-3 kg
$$
r_{\max}
=
\sqrt{
\frac{(3.9\times10^{-7})^2}
{16\pi(8.854\times10^{-12})(2.8\times10^{-3})(9.81)}
}
\approx
0.1115\,\mathrm{m}
$$

 

[anuttarasammyak]

I draw sketch of the configuration. Do I take it right ?

with no charge trasfer at the contact, which is done e.g., by coating insulator on sphere surface.
Does the critical E field strength come from insulation property?

 

[vela]

with no charge trasfer at the contact, which is done e.g., by coating insulator on sphere surface.

The sphere itself could be an insulator.

Does the critical E field strength come from insulation property?

It looks like the breakdown electric field strength for air.

 

[anuttarasammyak]

The sphere itself could be an insulator.

Thanks. Coating metal surface does not prevent sphere charges to accumulate near the contact by electrostatic induction.
The answer suggests e.g., homogeneous embedding of charges in insulator sphere.

 

[jnuz73hbn]

I draw sketch of the configuration. Do I take it right ?
View attachment 373529
with no charge trasfer at the contact, which is done e.g., by coating insulator on sphere surface.
Does the critical E field strength come from insulation property?

Ceiling = conducting and grounded (with a thin insulating surface layer).
Sphere = insulating spherical shell, with homogeneous fixed charge q=+3,8 \times10^{-7}\,\mathrm C.
No charge transfer between sphere and ceiling.
The method of image charges is to be considered.

 

[anuttarasammyak]

Thanks. Now all is clear to me to get the solution. How about you? What is the trouble you have?

 

[jnuz73hbn]

Thanks. Now all is clear to me to get the solution. How about you? What is the trouble?

Is it correct for the problem?
$$
r_{\max}
=
\sqrt{\frac{q^2}{16\pi\varepsilon_0mg}}
=
\sqrt{
\frac{(3.9\times10^{-7}\,\mathrm C)^2}
{16\pi(8.854\times10^{-12}\,\mathrm{F/m})(2.8\times10^{-3}\,\mathrm{kg})(9.81\,\mathrm{m/s^2})}
}
\approx
0.1115\,\mathrm m
=
11.15\,\mathrm{cm}
$$

$$
r_{\min}
=
\sqrt{\frac{|q|}{4\pi\varepsilon_0E_{\mathrm{crit}}}}
=
\sqrt{
\frac{3.9\times10^{-7}\,\mathrm C}
{4\pi(8.854\times10^{-12}\,\mathrm{F/m})(3.0\times10^6\,\mathrm{V/m})}
}
\approx
0.03418\,\mathrm m
=
3.42\,\mathrm{cm}
$$

 

[anuttarasammyak]

Which or both is your trouble getting these formula or these values from them?

 

[jnuz73hbn]

Which or both is your trouble getting these formula or these values from them?

My question is whether these formulas are sufficient to answer the question or if something is missing.

 

[anuttarasammyak]

if you have no troubles in derivation of these formula which give values such and such in calculation, I am afraid that I have nothing more to help you.