Explaining Why Total Charge Induced = -q for Image Problem

In summary, the classic image problem with a conducting plane and a point charge involves finding the charge density on the plane and then integrating to determine the total induced charge, which is equal to "-q" where q is the charge outside. However, it may not be immediately obvious why the induced charge is equal to the image charge. Some explanations include the cancellation of the electric field at distant points and the equipotential surface of the metal. Ultimately, image charges are not real and are used as a tool for understanding boundary value problems.
  • #1
adityatandon
3
0
Consider the classic image problem with a conducting plane and a point charge. After finding out the charge density on the plane we integrate to find out the total charge induced. It comes out to be "-q",where q is the charge outside-
My book says "It comes out to be -q, as you can convince yourself with the benefit of hindsight". It isn't that obvious why it should come out equal to the image charge. Can somebody explain?
 
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  • #2
adityatandon said:
Consider the classic image problem with a conducting plane and a point charge. After finding out the charge density on the plane we integrate to find out the total charge induced. It comes out to be "-q",where q is the charge outside-
My book says "It comes out to be -q, as you can convince yourself with the benefit of hindsight". It isn't that obvious why it should come out equal to the image charge. Can somebody explain?

At distant points the field due to the charge and its image cancel each other out. Now, since in fact the image is not there and the field is indeed caused by the induced charge, the sum of the induced charge and the point charge must be zero otherwise the field at distant points wouldn't be zero.
 
  • #3
I would describe it a little bit different.
For me, the main point is that the tangential component of the electric field on the plate has to vanish. Otherwise, there would be movement of the electrons at the surface of the metal. So, the metal's surface is an equipotential surface of the electrostatic potential. Now you can think of an imaginary charge distribution that could be found such that the potential vanishes on the surface of the metal if the metal was not there. Of course, you find that it is a symmetrically placed charge -q. You can go even further and generalize this procedure and ask for metallic corners subject to a point charge etc.

The bottom line is that image charges are not real, just something that gives us some intuition about boundary value problems :)
 
  • #4
Use Gauss's law. If the field is that from q'=-q, then that must also be the total surface charge.
 
  • #5


The total charge induced in the image problem is equal to -q because of the principle of charge conservation. In this problem, we have a point charge outside of a conducting plane, which creates an electric field that induces a charge on the plane. This induced charge must be equal and opposite to the point charge in order for the electric field to be canceled out inside the conductor.

When we integrate to find the total charge induced, we are essentially adding up all the small charges that make up the induced charge on the plane. Since the induced charge must be equal and opposite to the point charge, each small charge will have the same magnitude as the point charge, but with a negative sign.

Therefore, when we add up all these small negative charges, the total charge induced will be equal to -q, the same magnitude as the point charge outside. This is why it comes out to be -q, as stated in your book.

Additionally, the image charge technique is based on the concept of mirror charges, where an imaginary charge is placed on the opposite side of the conductor to mimic the behavior of the original charge. In this case, the image charge would also have a magnitude of -q, further supporting the result of the total charge induced being equal to -q.

In conclusion, the total charge induced being equal to -q in the image problem is a result of the principle of charge conservation and the concept of mirror charges. It may not seem obvious at first, but with a deeper understanding of these concepts, it becomes clear why this is the case.
 

1. Why does the total charge induced in an image problem equal -q?

In an image problem, a charge q is placed near a conducting surface. This charge q creates an electric field that is normal to the surface. The conducting surface then redistributes its charges to cancel out the electric field inside itself. This means that the induced charges create an electric field that is equal and opposite to the original electric field created by q. Since the total electric field inside the conducting surface must be zero, the total charge induced must also be equal and opposite to q, resulting in a total charge of -q.

2. How is the total charge induced calculated in an image problem?

The total charge induced in an image problem can be calculated using the principle of superposition. This means that the induced charges from each individual charge in the system can be added together to find the total induced charge. In the case of an image problem, the total induced charge is equal to the negative of the original charge, q.

3. Can the total charge induced ever be positive in an image problem?

No, the total charge induced in an image problem will always be negative. This is because the conducting surface will always redistribute its charges in a way that cancels out the electric field inside itself, resulting in a total charge of -q. Even if the original charge q is positive, the induced charges will still be negative to create an equal and opposite electric field.

4. How does the distance between the charge q and the conducting surface affect the total charge induced in an image problem?

The distance between the charge q and the conducting surface does not affect the total charge induced in an image problem. This is because the electric field created by q will always be normal to the conducting surface, and the conducting surface will always redistribute its charges to cancel out this electric field. The only factor that affects the total charge induced is the magnitude of the original charge q.

5. Why is it important to understand the concept of total charge induced in image problems?

Understanding the concept of total charge induced in image problems is important because it helps in understanding the behavior of electric fields near conducting surfaces. It also allows us to make predictions about the resulting electric field and potential near the conducting surface. This concept is also important in solving more complex electrostatic problems involving multiple charges and conducting surfaces.

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