Individual Spherical Capacitors

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Homework Help Overview

The discussion revolves around the capacitance of two conducting spheres with radii R1 and R2, which are initially separated. Participants explore the individual capacitance of the spheres and the implications of connecting them with a wire, questioning whether this connection results in a series or parallel configuration.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the individual capacitance formulas and express uncertainty about the configuration when the spheres are connected. Questions arise regarding the nature of series versus parallel connections and the implications of equal potential when connected.

Discussion Status

There is an ongoing exploration of the concepts of capacitance and the effects of connecting the spheres. Some participants provide insights into the definition of capacitance and the behavior of charges when the spheres are connected, while others seek further clarification on the implications of these connections.

Contextual Notes

Participants note that the problem does not explicitly state the initial potentials of the spheres and discuss the assumption that connecting them will equalize their potentials. There are references to the behavior of capacitors in series and parallel configurations, as well as the potential consequences of connecting the system to a battery.

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Homework Statement


Two conducting spheres of radii R1 and R2 are kept widely separated from each other. What are their individual capacitance? If the spheres are connected by a metal wire, what will be the capacitance of the combination? Think in terms of series-parallel connections.


Homework Equations





The Attempt at a Solution


Individual capacitance are simply [itex]4\pi\epsilon_oR_1[/itex] and [itex]4\pi\epsilon_oR_2[/itex]. I have no idea on what happens when they are connected. I can't understand the type of combination. Will it be series or parallel?

Any help is appreciated.
 
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A single spherical shell is a capacitor only if you imagine that the other shell (ground) is at infinitity. Now, when you connect the two spheres together, you join a set of plates of each of the capacitors together, and hold them at a fixed potential? What happens to the "other plates"? Is this connection in series or parallel?
 
Sorry but i have no idea, can you provide me some more explanation?
 
What does it mean for capacitors to be connected in series? How about in parallel?
 
Remember the definition of capacitance: C=Q/V. V is the potential with respect to infinity in this case. The shape does not count, any kind of a metal piece has capacitance.
Charge the spheres to equal potential. Q1=C1V, Q2=C2V. Connect them with a wire. What happens? What is the potential of this structure? What is the net charge on it? So what is the capacitance?

ehild
 
Dickfore said:
What does it mean for capacitors to be connected in series? How about in parallel?

If the capacitors are in series, the charge is same and if they are in parallel, the potential is same.

ehild said:
Remember the definition of capacitance: C=Q/V. V is the potential with respect to infinity in this case. The shape does not count, any kind of a metal piece has capacitance.
Charge the spheres to equal potential. Q1=C1V, Q2=C2V. Connect them with a wire. What happens? What is the potential of this structure? What is the net charge on it? So what is the capacitance?

ehild

Nothing should happen as they are at same potential.
 
And what is the charge of the whole system?

ehild
 
ehild said:
And what is the charge of the whole system?

ehild

The charge is (C1+C2)V.
And capacitance is (C1+C2).
 
Splendid! So it is like parallel capacitors.

ehild
 
  • #10
ehild said:
Splendid! So it is like parallel capacitors.

ehild

Yes, but in the question its not mentioned that they are at same potential.
 
  • #11
Well, if they are connected with a wire, they become the same potential anyway.

The capacitance does not depend on the charge or on the potential. But you can determine it by measuring charge and potential, and taking the ratio C=Q/V.

It is the same as with the mass. m=F/a, mass equals to force over acceleration. The mass of the object is its attribute, but you can measure it by applying a known force and detecting the acceleration.

If you know the charge of an object and its potential with respect to infinity you can attribute a capacitance to it as C=Q/V.

If your spheres are at different potential initially, after connecting them, there will be some charge transfer from one sphere to the other, till both of them have he same potential V=kQ1/R1=kQ2/R2. The first sphere is charged to Q1=VR1/k, the second one is charged to Q2=VR2/k. The resultant capacitance is (Q1+Q2)/V.

ehild
 
  • #12
ehild said:
Well, if they are connected with a wire, they become the same potential anyway.

The capacitance does not depend on the charge or on the potential. But you can determine it by measuring charge and potential, and taking the ratio C=Q/V.

It is the same as with the mass. m=F/a, mass equals to force over acceleration. The mass of the object is its attribute, but you can measure it by applying a known force and detecting the acceleration.

If you know the charge of an object and its potential with respect to infinity you can attribute a capacitance to it as C=Q/V.

If your spheres are at different potential initially, after connecting them, there will be some charge transfer from one sphere to the other, till both of them have he same potential V=kQ1/R1=kQ2/R2. The first sphere is charged to Q1=VR1/k, the second one is charged to Q2=VR2/k. The resultant capacitance is (Q1+Q2)/V.

ehild

Thank you very much for your explanation! :smile:

This is just a question out of curiosity.
What if i connected the whole system with the battery? Will they be in series or parallel?
 
  • #13
Pranav-Arora said:
What if i connected the whole system with the battery? Will they be in series or parallel?

What do you mean? Connect one of the spheres with one lead, and the other sphere with the other lead of the battery?
 
  • #14
Dickfore said:
What do you mean? Connect one of the spheres with one lead, and the other sphere with the other lead of the battery?

Oh yes, that's what i mean. :)
 
  • #15
Pranav-Arora said:
Oh yes, that's what i mean. :)

I told you. Each spherical capacitor has the other of its plates at infinity (and all these plates may be thought of as connected to the ground). If you draw the relevant diagram, you will immediately conclude if they are connected in series or parallel.
 
  • #16
See attachment. The connected spheres as capacitors are analogous to two "normal" grounded capacitor with the free plates connected. They are in parallel between B and the ground. What happens if you connect a battery in the way you said: one terminal to one plate of the first capacitor and the other terminal to the connected plate of the other capacitor?

See the connected spheres. They are of metal. So you make a metallic contact between the terminals of the battery. It is called "short", fatal for a battery. It gets very hot and starts to smoke and gets black and it is not a battery any more... RIP...

In case of a sphere as capacitor the other "plate" is at infinity. You need to connect the terminals of the battery to the wire connecting the spheres and to the other common point of the capacitors, which is at infinity.

ehild
 

Attachments

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  • #17
ehild said:
See attachment. The connected spheres as capacitors are analogous to two "normal" grounded capacitor with the free plates connected. They are in parallel between B and the ground. What happens if you connect a battery in the way you said: one terminal to one plate of the first capacitor and the other terminal to the connected plate of the other capacitor?

See the connected spheres. They are of metal. So you make a metallic contact between the terminals of the battery. It is called "short", fatal for a battery. It gets very hot and starts to smoke and gets black and it is not a battery any more... RIP...

In case of a sphere as capacitor the other "plate" is at infinity. You need to connect the terminals of the battery to the wire connecting the spheres and to the other common point of the capacitors, which is at infinity.

ehild

Thank you ehild for the explanation, it was an interesting read. :smile:
 

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