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Charge moving toward a conductor. Solution makes no physical sense!

  • #1
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?

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
ehild
Homework Helper
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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
 

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