Calculating Point Charge Potential at (0,0,0)

In summary, the problem of calculating the electrostatic potential at (0,0,0) for a point charge involves using the integral definition of the divergence to show that the divergence of r(vec)/r^3 is equal to 4pi delta(r), which can also be shown using the divergence theorem and taking the limit as r approaches 0. This concept is commonly found in graduate level textbooks.
  • #1
Yegor
147
1
Hallo!

Homework Statement


Consider electrostatic Potential of point charge at point (0,0,0)
[tex]\phi = 1/r[/tex]
I'm trying to calculate [tex]\Delta\phi[/tex]

Homework Equations



The Attempt at a Solution


Actually it's not a difficult problem outside (0,0,0):

[tex]\nabla\phi = -\frac{\vec r}{r^3}[/tex]
[tex]\Delta\phi = 0[/tex]

But i also know, that i should become Dirac-Delta function (charge density for point particle). What is the problem in point (0,0,0)?
I understand that Potential goes there to infinity, but how can i work it out mathematically?
 
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  • #2
Using the integral definition of div, it can be shown that
div(r(vec)/r^3)=4pi delta(r).
This can also be shown by appying the div theorem and taking the limit as r-->0.
What book are you using? Graduate texts do this.
 

1. What is the formula for calculating the point charge potential at (0,0,0)?

The formula for calculating the point charge potential at (0,0,0) is V(0,0,0) = kQ/r, where k is the Coulomb's constant, Q is the charge of the point charge, and r is the distance from the point charge to the point (0,0,0).

2. How do I determine the value of Coulomb's constant in the formula?

Coulomb's constant, denoted by k, has a value of 8.99x10^9 Nm^2/C^2. It is a fundamental constant in electromagnetism and can be found in most physics reference books or online resources.

3. Can the distance from the point charge to the point (0,0,0) be negative?

No, the distance r in the formula must be a positive value. If the point charge is located at (0,0,0), the distance will be 0 and the potential will also be 0.

4. How does the charge of the point charge affect the potential at (0,0,0)?

The potential at (0,0,0) is directly proportional to the charge of the point charge. This means that a larger charge will result in a larger potential, while a smaller charge will result in a smaller potential.

5. What are the units for the potential at (0,0,0)?

The units for the potential are joules per coulomb (J/C) or volts (V). This represents the amount of energy required to move a unit of charge from (0,0,0) to a point in space.

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