Solving Point Charge Impact Time on Grounded Plane

In summary, the conversation discusses a problem involving a point charge being released from rest at a certain distance from a grounded infinite conducting plane. The question is how long it takes for the charge to hit the plane. The answer is given as pi*(d/q)*sqrt(2pi*eps m d), but the process of getting to this answer involves solving a second order nonlinear equation and using the method of images. There is also a suggestion to use the substitution z = d cos2(theta) in order to solve for time.
  • #1
Euclid
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A point charge (q, mass m) is released from rest at a distance d from a grouned infinite conducting plane. How long does it take to hit the plane?
Answer pi*(d/q)*sqrt(2pi*eps m d)
This problem seemed easy to me at very, but it leads to a second order nonlinear equation
[tex]m\frac{d^2 z}{dt^2} = \frac{q^2}{16 \pi \epsilon_0 z^2}[/tex].

I tried using energy considerations to write v as v(z), I then solved for z as z(v), and integrated the above equation for v(t), putting in the limits 0 and infinity. This did not give the correct answer, although it appeared to be close. Any suggestions?
 
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  • #2
I haven't actually worked out the problem, so I'm not sure if the right side of your equation is correct. Just as a note incase you didn't do this - since the infinite conducting plane is grounded, you need to use the method of images to get your potential function.
 
  • #3
Maple choked on the d.e. but, you should be able to show
v2 = 2c(1/z - 1/d) with c = q2/(16 Pi Eps0 m)
with conservation of energy or integrating the force equation once.
now to solve for time put solve for v and use the positive root,
the negative one leads to t<0.
thus [tex] t = \frac{1}{\sqrt{2c}} \int_d^0 \sqrt{ \frac{ dz}{ d - z}} \. d z [/tex]
The substitution z = d cos2(theta) makes this doable.
 
  • #4
Yeah, that's what I indicated I tried above. It appeared to fail the first time I did it, but the second time it worked out.
Thanks!
 

1. What is the impact time of a point charge on a grounded plane?

The impact time of a point charge on a grounded plane refers to the amount of time it takes for the point charge to reach the grounded plane after being released from a certain distance.

2. How do you calculate the impact time of a point charge on a grounded plane?

The impact time can be calculated using the equation t = sqrt(2h/g), where t is the impact time, h is the initial height of the point charge, and g is the acceleration due to gravity.

3. What factors can affect the impact time of a point charge on a grounded plane?

The impact time of a point charge on a grounded plane can be affected by the initial height of the point charge, the charge of the point charge, and the strength of the electric field between the point charge and the grounded plane.

4. Can the impact time of a point charge on a grounded plane be negative?

No, the impact time of a point charge on a grounded plane cannot be negative as it represents a physical time measurement and cannot be less than zero.

5. How is the impact time of a point charge on a grounded plane important in scientific research?

The impact time of a point charge on a grounded plane is important in understanding the motion and behavior of charged particles in electric fields. It is also relevant in studying the effects of electric fields on materials and structures, and in designing electrical systems and devices.

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