Capacitance of a body versus capacitance of inscribed and circumscribed spheres

In summary, the capacitance of a conductive body is always between the capacitance of an inscribed sphere and the capacitance of a circumscribed sphere, as proven by the definition of capacitance and the relationship between capacitance and the size of a conductor. This has implications for the energy required to assemble a charge distribution, as the energy is also proportional to the size of the conductor and will therefore also be between the energy required for an inscribed sphere and a circumscribed sphere of the same potential or charge amount.
  • #1
bhh1988
4
0
just out of curiosity - it seems evident to me that the capacitance of a conductive body should be between the capacitance of an inscribed sphere and the capacitance of a circumscribed sphere. But I can't figure out how to prove this.

If you prove this, it follows that the energy required to assemble a charge distribution of some conductor is between the energy required to assemble a circumscribed sphere of the same potential and an inscribed sphere of the same potential (or of the same total amount of charge).
 
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  • #2
To prove this, start with the definition of capacitance. The capacitance of a single conductor is defined as the ratio of the charge on the conductor to the potential difference between its ends. For a given amount of charge, the potential difference will be larger for a larger conductor than it would be for a smaller conductor. In other words, the capacitance of a larger conductor is always greater than the capacitance of a smaller conductor.Now consider two conductors of different sizes, an inscribed sphere (with a radius r1) and a circumscribed sphere (with a radius r2). Since the capacitance is proportional to the size of the conductor, it follows that the capacitance of the inscribed sphere is always less than the capacitance of the circumscribed sphere. This proves that the capacitance of any conductor must be between the capacitance of an inscribed sphere and the capacitance of a circumscribed sphere.
 
  • #3


I find this question and its implications quite interesting. Capacitance is a measure of the ability of a body to store electrical charge, and it is affected by factors such as the geometry and material of the body. In the case of a conductive body, capacitance is also influenced by the amount of charge present on the body.

To understand the relationship between the capacitance of a conductive body and that of inscribed and circumscribed spheres, we first need to understand the concept of capacitance. Capacitance is defined as the ratio of the electric charge on a conductor to the potential difference between the conductor and its surroundings. In simpler terms, it is the measure of how much charge can be stored on a conductive body for a given potential difference.

Now, let's consider the capacitance of an inscribed sphere. We know that the capacitance of a sphere is proportional to its radius. As the radius of the sphere increases, so does its capacitance. This is because a larger sphere has a larger surface area, allowing for more charge to be stored on its surface.

On the other hand, the capacitance of a circumscribed sphere is inversely proportional to its radius. This is because the distance between the charges on the surface of the sphere and the surroundings increases as the radius increases, resulting in a decrease in the potential difference and thus the capacitance.

Now, if we consider a conductive body, its capacitance will depend on both its geometry and the amount of charge present. As the body is not a perfect sphere, its capacitance will fall somewhere between the capacitance of an inscribed sphere and a circumscribed sphere. This is because the charge distribution on the surface of the body will be a combination of the two - some charges will be closer to the surroundings, resulting in a lower potential difference and hence lower capacitance, while others will be closer to the center, resulting in a higher capacitance.

In terms of energy, the energy required to assemble a charge distribution on a conductive body will also fall between the energy required to assemble a circumscribed sphere and an inscribed sphere. This is because the energy required is directly proportional to the capacitance, and as we have seen, the capacitance of a conductive body is a combination of the capacitance of an inscribed and circumscribed sphere.

In conclusion, it is evident that the capacitance of a conductive body will fall between the
 

1. What is capacitance and how is it related to inscribed and circumscribed spheres?

Capacitance is a physical property that measures the ability of a body to store electrical energy. It is directly related to the geometry of the body, specifically the distance between two conductive surfaces and the permittivity of the material between them. Inscribed and circumscribed spheres refer to two different ways of enclosing a body, and they can affect the capacitance of the body by changing the distance between the conductive surfaces.

2. How does the capacitance of a body change when it is inscribed in a sphere?

When a body is inscribed in a sphere, the distance between the two conductive surfaces decreases. This results in an increase in capacitance, as the closer the surfaces are, the stronger the electric field and the more energy can be stored.

3. How does the capacitance of a body change when it is circumscribed by a sphere?

When a body is circumscribed by a sphere, the distance between the two conductive surfaces increases. This results in a decrease in capacitance, as the farther apart the surfaces are, the weaker the electric field and the less energy can be stored.

4. What is the significance of inscribed and circumscribed spheres in determining capacitance?

Inscribed and circumscribed spheres are important in determining capacitance because they represent two extremes of the distance between the conductive surfaces. By understanding how the capacitance changes in these extreme cases, we can better understand and calculate the capacitance of a body in more complex situations.

5. Is the capacitance of a body always greater when it is inscribed in a sphere compared to when it is circumscribed?

No, the capacitance of a body can be greater when it is circumscribed by a sphere, depending on the specific dimensions and materials involved. In general, however, the capacitance will increase when a body is inscribed in a sphere and decrease when it is circumscribed.

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