Derivation of the potential of a sphere

In summary, the potential used in the virial theorem for a constant density sphere is given by E=-\frac{3}{5}\frac{GM^2}{R}. This can be derived by considering the potential for a sphere of radius r, mass M created by bringing infinitly thin shells from infinity to form the next 'layer' of the sphere. The first line is a correction to the given result, while the second line is the sum of the potentials of spherical shells at a radius r and with a width of dr, assuming the potential is zero at infinity.
  • #1
Jayse_83
16
0
Hi,

I've been told in a lecture course that the potential used in the virial theorem (for our application) is 3/5 GM/r. This describes the potential for a sphere of radius r, mass M created by bringing infinitly thin shells from infinity to form the next 'layer' of the sphere. I am having difficulty deriving this for myself, anybody wana give it a try ??
 
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  • #2
For a constant density sphere, that's:

[tex]E=-\frac{3}{5}\frac{GM^2}{R}[/tex]

[tex]E=-\int_0^R \frac{GM_r}{r}dm[/tex]

[tex]M_r=\frac{4}{3}\pi r^3\rho[/tex]

[tex]dm=4\pi r^2\rho dr[/tex]

[tex]E=-\frac{16\pi^2G\rho^2}{3}\int_0^R r^4dr=-\frac{16\pi^2G\rho^2}{15}R^5[/tex]

[tex]M=\frac{4}{3}\pi R^3\rho[/tex]

[tex]E=-\frac{3}{5}\frac{GM^2}{R}[/tex]
 
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  • #3
Spacetiger: Where do the first two lines follow from?
 
  • #4
whozum said:
Spacetiger: Where do the first two lines follow from?

The first one is just correcting the result. I'm pretty sure he had mis-typed it. The second is summing over the potentials of spherical shells at a radius r and with a width of dr (brought in from infinity and assuming the potential is zero at infinity).
 
Last edited:
  • #5
yep i missed the squared out ... thanks for your help :)
 

Related to Derivation of the potential of a sphere

1. What is the potential of a sphere?

The potential of a sphere is the amount of work required to bring a unit positive charge from infinity to a point on the surface of the sphere, divided by the magnitude of the charge.

2. How is the potential of a sphere calculated?

The potential of a sphere can be calculated using the formula V = kQ/R, where k is the Coulomb's constant, Q is the charge on the sphere, and R is the radius of the sphere.

3. What are the units of potential of a sphere?

The units of potential are joules per coulomb (J/C) or volts (V).

4. How does the potential of a sphere change with distance?

The potential of a sphere decreases with distance from the sphere, following an inverse relationship. As the distance increases, the potential decreases.

5. What is the significance of the potential of a sphere?

The potential of a sphere is important in understanding the electric field and forces acting on charged particles around the sphere. It also plays a role in the behavior of conducting and insulating materials.

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