Electrostatics - two charged balls

[polibuda]

Homework Statement

Calculate the approximate value of the strength of the field on the surface of the metal sphere with radius r = 0.1 mm attached to a ball with radius R = 10cm and voltage U = 1kV.

Relevant Equations

E

Could somebody check my solution in this task? Is it correct?

 

[Delta2]

If the big ball is not from conducting material, and we neglect the surface charge density of the small metal sphere (due to electrostatic induction, probably we have the right to neglect it since the metal ball is very small), then your answer seems to be correct(as an approximation, but that's what the problem's statement asks for anyway)
However, if the big ball is from conducting material, then there would be charge flow between the two spheres, till they are at the same potential. The problem's solution becomes quite different in this case.

 

[polibuda]

I got to know, that both are from from conducting material. Well, what I should to do to solve task?

 

[haruspex]

I got to know, that both are from from conducting material. Well, what I should to do to solve task?

What will be the potential on the small sphere?

 

[polibuda]

What will be the potential on the small sphere?

Will it be the same like in big sphere?

 

[haruspex]

Will it be the same like in big sphere?

Yes.

 

[Delta2]

@haruspex this problem doesn't have an exact analytical solution right? To find an approximation (as the problem statement asks) is it right to consider the potential of each sphere equal to $k\frac{Q_i}{R_i}$ and then solve the system of equations $$K\frac{Q_1}{R_1}=K\frac{Q_2}{R_2}$$, $$Q_1+Q_2=Q_{TOT}$$

 

[haruspex]

@haruspex this problem doesn't have an exact analytical solution right? To find an approximation (as the problem statement asks) is it right to consider the potential of each sphere equal to $k\frac{Q_i}{R_i}$ and then solve the system of equations $$K\frac{Q_1}{R_1}=K\frac{Q_2}{R_2}$$, $$Q_1+Q_2=Q_{TOT}$$

Yes, but even simplify it a bit further. Just assume the smaller sphere goes to potential U and that this is almost entirely due to the charge it acquires. The charge on the larger sphere will have rearranged itself so as to create very little potential near the small sphere.

The field is more problematic. Where the two spheres face each other, their fields must cancel, right? And on the other side of the small sphere they will reinforce.
I think we have to interpret the question as asking for the field due to the charge on the small sphere, not the net field.

 

[Delta2]

Yes, but even simplify it a bit further. Just assume the smaller sphere goes to potential U and that this is almost entirely due to the charge it acquires.

Not sure I understand what simplification you mean. Do you mean that we should consider the final potential of both sphere, equal to the initial potential of the big sphere V=1KV?

The charge on the larger sphere will have rearranged itself so as to create very little potential near the small sphere.

Can you elaborate a bit more on that, cause i thought that the charge distribution on the big sphere will stay almost unaffected because very small charge will move into the smaller sphere.

 

[haruspex]

Do you mean that we should consider the final potential of both sphere, equal to the initial potential of the big sphere V=1KV?

Yes.

i thought that the charge distribution on the big sphere will stay almost unaffected because very small charge will move into the smaller sphere.

It won't be much charge, but because of the small radius it will be a significant field.
Don't your equations in post #7 lead to that?

Edit: But now I think I am being inconsistent, having argued for equal and opposite fields.

 

[polibuda]

@haruspex this problem doesn't have an exact analytical solution right? To find an approximation (as the problem statement asks) is it right to consider the potential of each sphere equal to $k\frac{Q_i}{R_i}$ and then solve the system of equations $$K\frac{Q_1}{R_1}=K\frac{Q_2}{R_2}$$, $$Q_1+Q_2=Q_{TOT}$$

It is my new solution, but I think it is completely bad done. I have the qeustions to equations. Is the first electric potential (V1) equal to 10kV and the second equal to K*(charge of small sphere)/(radius of small sphere)? Is total charge the sum of charge of big sphere (Q1) and charge of small sphere (Q2)?

 

[Delta2]

You went the wrong direction when you setup this equation $$U=V_1+V_2$$. The correct equation to use is $V_1=V_2$.

U=1KV is useful only to determine the total initial charge which i think you calculate correctly as $$Q=1,11\cdot10^{-8}$$

 

[polibuda]

You went the wrong direction when you setup this equation $$U=V_1+V_2$$. The correct equation to use is $V_1=V_2$.

U=1KV is useful only to determine the total initial charge which i think you calculate correctly as $$Q=1,11\cdot10^{-8}$$

Thank you, but I have problem with understanding equations.
$V_1 = 10kV ?$
$V_2 = K*(Charge Of Small Sphere)/(Radius Of Small Sphere) ?$

 

[Delta2]

Thank you, but I have problem with understanding equations.
$V_1 = 10kV ?$
$V_2 = K*(Charge Of Small Sphere)/(Radius Of Small Sphere) ?$

$V_2$ is correct.
Why do you think $V_1$ is 10kV?

 

[Delta2]

$V_1$ is the potential that the big sphere will have after the charge flow has ended. In your work in post #11 you set it equal to $K\frac{Q-Q_2}{R}$ which is correct.

 

[polibuda]

$V_2$ is correct.
Why do you think $V_1$ is 10kV?

Beacuse it is given as voltage in task.

 

[polibuda]

$V_1$ is the potential that the big sphere will have after the charge flow has ended. In your work in post #11 you set it equal to $K\frac{Q-Q_2}{R}$ which is correct.

Ahhh, ok. Thank you very much for help. Now, I should cope with solving task.