Potential generated by a point charge in a isotropic medium

In summary, the charge of the point particle can be found by taking the limit of the charge distribution as the radius approaches 0, which yields q = -4πε0A. This is consistent with the fact that the point charge is located at the origin.
  • #1
gabu
5
0

Homework Statement



When a point charge is positioned at the origin = 0 in an isotropic
material, a separation of charge occurs around it, the Coulomb field of the
point charge is screened, and the electrostatic potential takes the form

[itex] \phi(r) = \frac{A}{r} \exp\left( -\frac{r}{\lambda} \right)[/itex]

Here, r is the distance from the origin, and A and are constants. In this case,
determine the charge density distribution () in the space surrounding the
origin.

(b)
In formula (1) above, the potential becomes infinitely large at the origin. This
shows that the point charge is located at the origin. Using Gauss’s Law, find
the charge of the point particle.

Homework Equations


[/B]
Electric Potential
[itex] \phi(r) = \frac{A}{r} \exp\left( -\frac{r}{\lambda} \right)[/itex]

Laplacian in spherical coordinates
[itex] \Delta = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2\,\frac{\partial}{\partial r}\right)[/itex]

The Attempt at a Solution



To solve the first part, I have used Poisson's equation [itex] \Delta \phi(r) = -\frac{\rho}{\epsilon_{0}}[/itex]

and obtained

[itex] \rho = \frac{A\epsilon_{0}}{r\,\lambda^2}\,\exp\left( -\frac{r}{\lambda}\right) [/itex]

for the charge distribution surrounding the origin. For the second part of the problem I used the fact that the potential is spherically symmetrical to write Gauss' law as

[itex] \vec{\nabla}\phi\cdot \hat{r}\int dA = \frac{Q}{\epsilon_{0}}[/itex]
[itex] \vec{\nabla}\phi\cdot \hat{r}(4\pi\epsilon_{0}) = \frac{Q}{\epsilon_{0}}[/itex]
finally obtaining

[itex] q = -4\pi\epsilon_{0}A\left(1-\frac{r^2}{\lambda}\right) \,\exp\left( -\frac{r}{\lambda}\right) [/itex]

My problem with this solution is that it depends on the distance from the origin, but the problem said it was a point charge. The only thing I can think about to go around this is considering r=0, but I don't have a good justification for this. What did I get wrong?

Thank you very much.
 
Physics news on Phys.org
  • #2
You are computing the charge within a radius ##r##. This includes both the point charge at the origin and any charge from the continuous distribution you computed in (a) up to the radius ##r##. In order to get the point charge only, you need to take the limit ##r\to 0##.
 

1. What is the definition of potential generated by a point charge in an isotropic medium?

The potential generated by a point charge in an isotropic medium refers to the amount of electric potential energy per unit charge at any given point in the medium, caused by the presence of a point charge. It is a measure of the strength of the electric field at that point.

2. How is the potential generated by a point charge in an isotropic medium calculated?

The potential generated by a point charge in an isotropic medium can be calculated using the equation V = kq/r, where V is the potential, k is the Coulomb's constant, q is the magnitude of the point charge, and r is the distance from the point charge to the point where the potential is being measured.

3. What is the unit of measurement for potential generated by a point charge in an isotropic medium?

The unit of measurement for potential generated by a point charge in an isotropic medium is volts (V). This unit is equivalent to joules per coulomb (J/C).

4. How does the potential generated by a point charge in an isotropic medium vary with distance?

The potential generated by a point charge in an isotropic medium follows an inverse relationship with distance. This means that as the distance from the point charge increases, the potential decreases. This relationship is described by the equation V ∝ 1/r, where V is the potential and r is the distance.

5. What is the significance of an isotropic medium in relation to the potential generated by a point charge?

An isotropic medium is a medium that has the same properties in all directions. In the context of potential generated by a point charge, this means that the potential at any given point is the same regardless of the direction in which it is measured. This is because the point charge is assumed to be equally distributed in all directions, resulting in a uniform electric field and potential in an isotropic medium.

Similar threads

  • Introductory Physics Homework Help
2
Replies
64
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
17
Views
309
  • Introductory Physics Homework Help
Replies
1
Views
809
  • Introductory Physics Homework Help
Replies
4
Views
855
  • Introductory Physics Homework Help
Replies
11
Views
623
Replies
4
Views
331
  • Introductory Physics Homework Help
Replies
3
Views
761
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
639
Back
Top