Energy stored in a charged sphere

In summary, the conversation discusses the use of the equation for calculating the energy stored in a charged sphere and determining the potential function that should be used. It is stated that the electrostatic potential should be used, as it is the solution to the Legendre equation. The two given equations are shown to deliver the same result and the constant in the potential function can be determined by setting the energy to zero for a homogeneously charged sphere. The exact solution can also be found by solving the Legendre equation in spherical coordinates.
  • #1
Kosta1234
46
1
Homework Statement
Energy stored in a charged sphere
Relevant Equations
$$ U = \frac {1}{2} \cdot \int \phi (r) \cdot \rho(r) dV $$
Hi.
When I am asked to figure out the Energy stored in a charged sphere and I use this equation: ## U = \frac {1}{2} \cdot \int \phi (r) \cdot \rho(r) dV ##
what is the potential ## \phi ( r) ## stands for? I tried to use the potential inside the sphere, because out side of the sphere ## \rho (r) = 0 ##, and I tried to sum those to up.
I'm not getting the same answer as in this equation:
$$ U = \frac {\varepsilon }{2} \cdot \int_{all space} E^2 dV $$so what ## \phi (r) ## I've to use and why?Edit: I got it right, I think.
was I right when I said to use the potential inside the sphere, because out side of the sphere ## \rho (r) = 0 ##
and to use the ## U = \frac {\varepsilon }{2} \cdot \int_{all space} E^2 dV ## to all space?
 
Last edited:
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  • #2
It's the electrostatic potential you should use, i.e., the solution of the Legendre equation,
$$\Delta \phi(\vec{x})=-\frac{1}{\epsilon} \rho(\vec{x}).$$
This potential is defined up to a constant, and so is ##U##.

It's simple to show that (up to a constant) your two expressions deliver the same result. To see this, just use
$$\vec{E}=-\vec{\nabla} \phi$$
in one of the factors in the integral
$$\tilde{U}=\frac{\epsilon}{2} \int_{\mathbb{R}^3} \mathrm{d}^3 x \vec{E}^2(\vec{x}),$$
and then Gauss's theorem for 3D partial integration! It's a good exercise!

BTW: You can fix the constant by demanding that ##U=0## for ##\rho=Q/V## (where ##Q## is the total charge, and ##V## the volume of the sphere; I guess you mean a homogeneously charged sphere).

You can also find the exact solution of this problem by solving the Legendre equation in shperical coordinates.
 

Related to Energy stored in a charged sphere

1. What is the formula for calculating the energy stored in a charged sphere?

The formula for calculating the energy stored in a charged sphere is E = (1/2) * (Q^2 / 4πε₀R), where E is the energy, Q is the charge, R is the radius of the sphere, and ε₀ is the permittivity of free space.

2. How does the energy stored in a charged sphere change with increasing charge?

The energy stored in a charged sphere increases proportionally with the square of the charge. This means that as the charge increases, the energy stored also increases, but at a faster rate.

3. What factors affect the energy stored in a charged sphere?

The energy stored in a charged sphere is affected by the charge of the sphere, the radius of the sphere, and the permittivity of free space. Additionally, the presence of other charged objects in the surrounding area can also affect the energy stored.

4. How is the energy stored in a charged sphere related to the electric potential?

The energy stored in a charged sphere is directly related to the electric potential. The higher the electric potential, the more energy is stored in the sphere. This relationship is described by the formula E = QV, where E is the energy, Q is the charge, and V is the electric potential.

5. Can the energy stored in a charged sphere be negative?

Yes, the energy stored in a charged sphere can be negative. This occurs when the sphere is negatively charged and is in an electric field that is directed towards the sphere. In this case, the energy stored is negative because the electric field is doing work on the sphere, reducing its energy.

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