Charge moving toward a conductor. Solution makes no physical sense

In summary, the problem involves a point charge q of rest mass m being released from rest a distance d from an infinite grounded conducting plane. The solution is given as (πd/q) √(2πϵ_0 md) and Griffiths suggests using the method of images. The differential equation for the problem yields a solution of r as a function of time being some constant times t to the 2/3 power, but this contradicts the physical expectation that the distance between the charge and plane decreases as time goes forward. The error is found to be a missing minus sign in the equation, as the mirror charge should be opposite to the real one, resulting in a force pointing inward towards the plane.
  • #1
thatsunpossibl
1
0
This is not an assigned homework problem, just something I came across. It's in Griffiths E&M book.
Also I apologize if my equations don't look right. They are showing up all weird on my computer screen.

Homework Statement



A point charge q of rest mass mass m is released from rest a distance d from an infinite grounded conducting plane. How long will it take for the charge to hit the plane?

Homework Equations



F=k (q_1 q_2)/r^2 =m[r]\ddot{}[/itex]


The Attempt at a Solution


The solution is given as

(πd/q) √(2πϵ_0 md)

Griffth implies that we should use the method of images, comparing the problem to two oppositely charged particles moving toward each other. I can get the solution by solving the differential equation above. In fact, mathematically the only solution to that is that r as a function of time comes out to r being some constants times t to the 2/3 power. The problem is that it doesn't seem to be a real physical solution! It implies that r gets bigger as time goes forward, which is not the case. also, if you then differentiate that equation to find v as a function of t, we'll get v as some constants times t to the -1/3 power. This implies that v decays as time goes forward, which can't be true. The charge has to be gaining speed as it falls right? Same with acceleration. Shouldn't it be gaining acceleration as it gets closer to the sheet?



 
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  • #2
thatsunpossibl said:
This is not an assigned homework problem, just something I came across. It's in Griffiths E&M book.
Also I apologize if my equations don't look right. They are showing up all weird on my computer screen.

Homework Statement



A point charge q of rest mass mass m is released from rest a distance d from an infinite grounded conducting plane. How long will it take for the charge to hit the plane?

Homework Equations



F=k (q_1 q_2)/r^2 =m[itex]\ddot{r}[/itex]

ehild


The Attempt at a Solution


The solution is given as

(πd/q) √(2πϵ_0 md)

Griffth implies that we should use the method of images, comparing the problem to two oppositely charged particles moving toward each other. I can get the solution by solving the differential equation above. In fact, mathematically the only solution to that is that r as a function of time comes out to r being some constants times t to the 2/3 power. The problem is that it doesn't seem to be a real physical solution! It implies that r gets bigger as time goes forward, which is not the case. also, if you then differentiate that equation to find v as a function of t, we'll get v as some constants times t to the -1/3 power. This implies that v decays as time goes forward, which can't be true. The charge has to be gaining speed as it falls right? Same with acceleration. Shouldn't it be gaining acceleration as it gets closer to the sheet?

A minus sign is missing from the equation: The mirror charge is opposite to the real one, so the force points inward, towards the plane.


ehild
 

1. How does charge moving toward a conductor affect the conductor's properties?

When charge moves toward a conductor, it creates an electric field that induces a flow of electrons within the conductor. This flow of electrons causes the conductor's properties, such as resistance and capacitance, to change.

2. Why does the solution of charge moving toward a conductor sometimes make no physical sense?

This can happen when the charge is moving at a high velocity or when the conductor's properties are not considered. In these cases, the equations used to calculate the solution may not accurately depict the real-world scenario.

3. Can charge moving toward a conductor cause damage?

Yes, in certain situations, charge moving toward a conductor can cause damage. For example, if the charge is moving at a high velocity, it can generate a large amount of heat in the conductor, potentially damaging it.

4. How does the shape of the conductor affect the behavior of charge moving toward it?

The shape of the conductor can have a significant impact on the behavior of charge moving toward it. For example, sharp edges can create areas of high electric field intensity, which can affect the flow of charge and potentially cause damage.

5. What are some factors that can affect the accuracy of the solution for charge moving toward a conductor?

Some factors that can affect the accuracy of the solution include the velocity of the charge, the properties of the conductor, and the presence of other nearby conductors or insulators. Additionally, the assumptions and simplifications made in the equations used to calculate the solution can also impact its accuracy.

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