# Derivation of the potential of a sphere

• Jayse_83
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.
Jayse_83
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 ??

For a constant density sphere, that's:

$$E=-\frac{3}{5}\frac{GM^2}{R}$$

$$E=-\int_0^R \frac{GM_r}{r}dm$$

$$M_r=\frac{4}{3}\pi r^3\rho$$

$$dm=4\pi r^2\rho dr$$

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

$$M=\frac{4}{3}\pi R^3\rho$$

$$E=-\frac{3}{5}\frac{GM^2}{R}$$

Last edited:
Spacetiger: Where do the first two lines follow from?

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:
yep i missed the squared out ... thanks for your help :)

## 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.

• Introductory Physics Homework Help
Replies
43
Views
2K
• Advanced Physics Homework Help
Replies
2
Views
2K
• Introductory Physics Homework Help
Replies
16
Views
589
• Introductory Physics Homework Help
Replies
21
Views
2K
• Introductory Physics Homework Help
Replies
23
Views
422
• Introductory Physics Homework Help
Replies
10
Views
523
• Introductory Physics Homework Help
Replies
14
Views
2K
• Quantum Physics
Replies
2
Views
540
• Introductory Physics Homework Help
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
1K