Why does the method of images use a distance of 2x instead of just x?

In summary, the student is trying to find the electric field close to the metal surface as a function of distance from the rod.
  • #1
-Vitaly-
39
0

Homework Statement


A point charge +q is initially at distance x from a conducting plane of infinite extent and held at zero potential. Find the work done in moving the charge to an infinite distance from the plane. Hence find the minimum energy an electron must have in order to escape
from a metal surface (assume that it starts at a distance 0.1nm, which is about one atomic diameter, from it). Express your answer in electron-volts.
[Answers: q^2/(16πε0x) ; 3.6eV]

Homework Equations


U=q(V2-V1)
Method of images

The Attempt at a Solution


The plane can be replaced with a negative -q point charge at a distance 2x. Therefore the energy required to move the charge to infinity=q(0-(-q/(4πε02x)))=q^2/(8πε0x). Why did they put 16 in the answer??
 
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  • #2
The reason for the factor of 1/2 is that the potential you've calculated is only valid above the plane; so if you think of the energy stored in the fields you see that if it really were just two point charges, the energy would be the same above and below the plane. In this case, the presence of the conductor means that only half of that energy is present---the half above the conductor.

You can work this out explicitly using two different methods:

(1) [tex]W=\frac{\epsilon_0}{2} \int_{\text{all space}}E^2dV[/tex]

--keep in mind that the field is actually zero below the plane!

(2)[tex]W=\int_{x}^{\infty} \vec{F}\cdot\vec{dl}=q\int_{x}^{\infty} \vec{E}\cdot\vec{dl}[/tex]
 
  • #3
Ok, thank you :)
 
  • #4
Another problem I've got:

Homework Statement


An infinite, thin, uniformly charged rod (line charge density λ) is situated parallel to a
metal plate a distance d above it. Use E=λ/(2πε0x ) (E at a point x m away from the rod, no other conductors are present) to calculate the E-field close
to the surface of the plate as a function of perpendicular distance to the rod.

Homework Equations


E=λ/(2πε0x

The Attempt at a Solution


Well, I drew field lines of the rod and the plane, and if looked in a plane perpendicular to the rod, the field lines look like half of a dipole, so maybe I thought maybe I can treat it as a dipole in this plane with charge dq? But I'm not sure what to do next? just add the two E?
http://img22.imageshack.us/img22/1980/clipboard01ro5.jpg [Broken]
Thanks
 
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1. Why is the distance in the method of images double that of the actual distance?

The method of images uses a distance of 2x instead of just x because it allows for a more accurate representation of the electric field. This distance represents the distance between the two charges in the image charge method, creating a symmetrical distribution of charges that accurately reflects the real electric field.

2. How does using a distance of 2x affect the accuracy of the method of images?

The use of a distance of 2x in the method of images improves the accuracy of the method by accounting for the effects of the real charge on the electric field. It creates a more symmetrical distribution of charges, resulting in a more accurate representation of the electric field.

3. Why is the distance in the method of images always twice that of the actual distance?

The distance in the method of images is always twice that of the actual distance because it is based on the principle of reflection. The image charge is a reflection of the real charge, and the distance between them is equal to the distance between the real charge and its image, which is twice the actual distance.

4. Is there a specific reason why the distance in the method of images is double that of the actual distance?

Yes, the use of a distance of 2x in the method of images is based on the principle of symmetry. It ensures that the electric field is symmetrical and accurately reflects the real charge, resulting in a more precise calculation of the electric field at any given point.

5. Can the distance used in the method of images be changed to a value other than 2x?

Technically, yes, the distance used in the method of images can be changed to a value other than 2x. However, using a distance other than 2x would not accurately represent the electric field and would result in a less precise calculation. Therefore, a distance of 2x is the most commonly used and recommended value for the method of images.

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