Principle of superposition for charges/induced charges

In summary, the key observation to solve the above problem is that the charge Q can be dragged out into a flat capacitor plate parallel to the 2 existing plates. Apparently, while the charge distribution on the 2 existing plates changes, the total charge induced on each plate remains the same, due to the principle of superposition.
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phantomvommand
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The key observation to solve the above problem is that the charge Q can be dragged out into a flat capacitor plate parallel to the 2 existing plates. Apparently, while the charge distribution on the 2 existing plates changes, the total charge induced on each plate remains the same, due to the principle of superposition.

How and why does the principle of superposition work here??

If I had 2 concentric spherical shells instead of 2 parallel plates, and I placed a charge in the region between the concentric shells, can I likewise drag the charge out into a shell and claim that the total induced charges on each of the 2 existing shells remains the same due to the principle of superposition?

If the answer to the above question is yes, when does the principle of superposition fail? Please do give some examples, that are ideally somewhat similar to the above questions, so as to elucidate the exact criterions that cause the principle of superposition to be inapplicable.

Thank you!
 
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  • #2
There are a few things going on here!

Firstly, take a capacitor with plates of infinite extent (say, each parallel to the ##xy## plane). By translational symmetry parallel to the plates it follows the charges ##q_A## and ##q_B## induced on the capacitor plates will not depend on the coordinates ##(x,y)## at which you position an arbitrary charge ##q##. For this problem, then, it's necessary to assume that if ##A \propto l^2##, then ##l \gg d##, or in other words that edge effects are negligible.

Now for the superposition part. Suppose you have a set of charges ##\{ Q^{(1)}, \dots, Q^{(N)} \}##. If you put only the charge ##Q^{(1)}## between the plates (at the specified ##z## value, and arbitrary ##(x,y)## coordinates), then the corresponding charge induced on the plates will be ##q_A^{(1)}## and ##q_B^{(1)}##. But by superposition, you know that the total charges induced on the plates, ##q_A## and ##q_{B}##, will just be the sum of those that would be induced if the ##Q^{(i)}## charges were on their own one at a time, i.e.$$q_{A} = q_A^{(1)} + \dots + q_A^{(N)}, \quad \quad q_{B} = q_B^{(1)} + \dots + q_B^{(N)}$$So the solution to the problem is to note that you can start with a charge ##Q##, partition it into ##N## smaller charges and spread them out parallel to the plates to different locations, without affecting the total charge on either plates. Further, by Gauss, you can write down$$q_{A} + q_{B} + Q = 0$$To work out the ratio ##q_{A}/q_{B}##, now just use the fact that you have effectively two capacitors in series, with the two outermost plates at the same electric potential!
 
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  • #3
etotheipi said:
There are a few things going on here!

Firstly, take a capacitor with plates of infinite extent (say, each parallel to the ##xy## plane). By translational symmetry parallel to the plates it follows the charges ##q_A## and ##q_B## induced on the capacitor plates will not depend on the coordinates ##(x,y)## at which you position an arbitrary charge ##q##. For this problem, then, it's necessary to assume that if ##A \propto l^2##, then ##l \gg d##, or in other words that edge effects are negligible.

Now for the superposition part. Suppose you have a set of charges ##\{ Q^{(1)}, \dots, Q^{(N)} \}##. If you put only the charge ##Q^{(1)}## between the plates (at the specified ##z## value, and arbitrary ##(x,y)## coordinates), then the corresponding charge induced on the plates will be ##q_A^{(1)}## and ##q_B^{(1)}##. But by superposition, you know that the total charges induced on the plates, ##q_A## and ##q_{B}##, will just be the sum of those that would be induced if the ##Q^{(i)}## charges were on their own one at a time, i.e.$$q_{A} = q_A^{(1)} + \dots + q_A^{(N)}, \quad \quad q_{B} = q_B^{(1)} + \dots + q_B^{(N)}$$So the solution to the problem is to note that you can start with a charge ##Q##, partition it into ##N## smaller charges and spread them out parallel to the plates to different locations, without affecting the total charge on either plates. Further, by Gauss, you can write down$$q_{A} + q_{B} = Q$$To work out the ratio ##q_{A}/q_{B}##, now just use the fact that you have effectively two capacitors in series, with the two outermost plates at the same electric potential!

This has been very helpful, thank you very much!
 
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What is the principle of superposition for charges/induced charges?

The principle of superposition for charges/induced charges states that the total electric field at a point in space is equal to the vector sum of the electric fields produced by each individual charge present in that space. This means that the electric field at a point is the sum of the electric fields produced by all the charges present, and it does not depend on the order in which the charges were placed.

How does the principle of superposition for charges/induced charges apply to electric fields?

The principle of superposition for charges/induced charges applies to electric fields by allowing us to calculate the total electric field at a point in space by adding the electric fields produced by each individual charge present. This is a useful tool in understanding the behavior of electric fields and their interactions with charges.

Can the principle of superposition for charges/induced charges be used to calculate the electric field at any point?

Yes, the principle of superposition for charges/induced charges can be used to calculate the electric field at any point in space. As long as we know the charges present and their positions, we can use the principle of superposition to find the total electric field at a given point.

What is the difference between superposition for charges and superposition for induced charges?

The principle of superposition for charges refers to the total electric field produced by multiple charges present in a space, while the principle of superposition for induced charges refers to the total electric field produced by charges that are induced on a conductor due to the presence of an external electric field. In other words, the former involves charges that are already present, while the latter involves charges that are induced by an external influence.

How is the principle of superposition for charges/induced charges related to Coulomb's law?

The principle of superposition for charges/induced charges is related to Coulomb's law in that both involve the concept of superposition. While Coulomb's law describes the force between two charges, the principle of superposition expands this concept to multiple charges, allowing us to calculate the total electric field at a point due to all the charges present.

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