Limitations of the Image Method

[jamesEsnow]

Hi everybody, new here. Okay, I will get straight to the chase. I am an undergrad student and I am current taking a course in electromagnetics--transmissions lines...

Anyways we just had an exam yesterday and there was a question on it that I believe may have serious flaws. I will try to explain it as well as possible without diagrams. If I can't, please let me know and I will draw up a pic and post.

So, the question stated that there were three point charges and a conducting surface on the x-y plane. Now, two of these charges will be positive and equal in magnitude: Charge 1 at (0,1,1) and Charge 2 at (0,-1,1). Here is the kicker. The third charge is located at (0,0,0) and has a positive charge of some other magnitude (can't) remember. The question asked for the force on the third charge at (0,0,0.)

By pure accident I did not see there was a conducting plane and I went ahead and used Coulombs law for the electric fied at the origin--easy stuff. On reviewing my answers I saw that a conducting plane was there. I initially thought of the method of images but it was not stated whether the conducting surface was perfect and it was NOT grounded. Furthermore, the material wasn't stated either.

Of course using image theory is easy for point charges and I understand that many problems in class are purely academic. I also understand that it would probably be reasonable to ask for an instantaneous force on the charge. In retrospect, I believe that the instructor just wanted us to use image theory anyways. However, at the time I did not use image theory and was quite confused: 1. I have never seen a problem where a charge was placed on the conducting surface in an image theory application. 2. From what I understand, the conductor must be perfect and/or grounded. 3. I have read that most metals can be approximated and material was not listed. 4. I assumed the charge will dissipate in the conductor, so it couldn't be measured with image method.

I peered further into the text and I found a section on free-charge dissipation. There was an equation that states that the charge density as a function of time equals intial charge density times e^-(σ/ε)t. Now if the conductor is perfect, then wouldn't the charge dissipate virtually instantaneously. And doesn't image method work because of "mirrored" charges moving in the conductor to counter the charge above the plane. So, it seems like by the time image method was applicable, the placed charge would have already dissapated. Therefore, the only measurement you can take is instantaneous, and this should be close to Coulomb's Law.

Now, I will cop up and accept the missed points if I was wrong, but there seems to be something wrong here. Please give me some insight. Thank You.

 

[anathema]

Dear fellow student,

Thank you for bringing this question to our attention. it is always important to question and analyze information presented to us, especially in exams and assignments. After reading your post, I can understand your confusion and frustration with this particular question. I will try my best to provide some insight and clarify any misunderstandings.

Firstly, your understanding of the method of images is correct. It is a commonly used technique in solving problems involving conducting surfaces, and it does require the surface to be perfect and grounded. However, in some cases, an imperfect or ungrounded conductor can still be approximated and the method of images can still be applied with some modifications. It really depends on the specific problem and the material properties of the conductor. In this case, since the material was not specified, it is safe to assume that the conducting surface can be approximated and the method of images can be used.

Secondly, your concern about the charge dissipating in the conductor is valid. However, in this scenario, the charge on the conducting surface is not moving or changing, so the equation for free-charge dissipation does not apply. The image charge is simply a mathematical construct to help solve the problem, and it does not actually exist in the conductor. Therefore, the charge at the origin will not dissipate and the method of images can still be used to find the force on it.

Lastly, I would like to address your statement about the only measurement being instantaneous and close to Coulomb's law. While it is true that the method of images is used to find the instantaneous force, it does not necessarily mean that it will be close to Coulomb's law. The method of images takes into account the presence of the conducting surface and its effect on the electric field, which can result in a different force than what Coulomb's law predicts. This is why the method of images is used in these types of problems, to provide a more accurate solution.

In conclusion, while your initial approach to the question may have been correct, using the method of images is the preferred method in this scenario. I hope this helps clarify any confusion and I wish you the best of luck in your studies. Remember to always question and analyze information presented to you and don't be afraid to seek clarification when needed. That is the essence of being a scientist.

 

[m2003]

I understand your concerns and confusion regarding the limitations of the image method in this specific scenario. First of all, it is important to note that the image method is a mathematical technique used to simplify the solution of certain electrostatics problems, and it is not a physical phenomenon. Therefore, it is important to carefully consider the assumptions and conditions under which the image method can be applied.

In this case, as you have correctly pointed out, the conducting surface must be perfect and/or grounded for the image method to be applicable. Additionally, the material of the conductor must be known in order to accurately calculate the charge dissipation. These are important factors that were not specified in the problem, which can lead to incorrect solutions.

Furthermore, as you have mentioned, the image method is based on the concept of mirrored charges in the conductor. However, in this scenario, the third charge is placed directly on the conductor which may disrupt the charge distribution and make the use of the image method inaccurate.

It is also worth noting that the image method is most applicable for steady-state problems, where the charge distribution has reached equilibrium. In this case, the problem asks for the force on the third charge at a specific moment in time, which may not be accurately represented by the image method.

In conclusion, while the image method is a useful tool in certain electrostatics problems, it is important to carefully consider the conditions and assumptions under which it can be applied. In this scenario, it may be more appropriate to use Coulomb's law to calculate the force on the third charge at the given moment in time. Thank you for raising this concern and seeking further clarification.