Purcell's Electricity and Magnetism - question

[apw235]

Homework Statement

Purcell 3.3: In the field of a point charge over a plane, if you follow a field line that starts at the point charge in a horizontal direction, that is, parallel to the plane, where does it meet the surface of the conductor?

Homework Equations

The problem 'hint' is "You'll need Gauss' law and a simple integration."

The Attempt at a Solution

The electric field on the surface of the conductor at a radius R=$$\sqrt{r^{2}+h^{2}}$$ (h is the height of the pt charge, r is the x component of the radius on the plane), the Electric field due to the point charge is:
E=$$\frac{-2Qh}{(r^{2}+h^{2})^{3/2}}$$. This is given in the book.
I have no idea where to start..

 

[apw235]

anybody?

 

[Moetasim]

I would first clarify the question and the given information. It appears that the homework statement is asking for the point of intersection between a field line from a point charge and a plane. The given information includes the electric field at the surface of the conductor and a hint to use Gauss' law and integration.

To solve this problem, I would start by drawing a diagram to better visualize the situation. From the given information, we know that the electric field on the surface of the conductor is given by E=\frac{-2Qh}{(r^{2}+h^{2})^{3/2}}. This means that the field lines would be radial, with the direction of the field pointing away from the point charge.

Next, I would use Gauss' law to determine the total electric flux through a closed surface surrounding the point charge. This can be done by using a Gaussian surface in the shape of a cylinder with one end on the surface of the conductor and the other end enclosing the point charge. The electric flux through this surface would be Q, the charge enclosed by the surface.

Since the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space, we can set this equal to the total electric flux calculated using the electric field equation given in the problem. This will give us an equation in terms of the distance from the point charge to the surface of the conductor, r.

Solving for r, we can find the distance at which the field line intersects the surface of the conductor. This would be the point where the field line from the point charge becomes parallel to the surface of the conductor.

Finally, to determine the exact point of intersection, we can use integration to find the distance along the field line from the point charge to the surface of the conductor. This would involve integrating the electric field equation from the point charge to the point of intersection, and setting it equal to the total electric flux calculated using Gauss' law.

In summary, to solve this problem, we would use Gauss' law and integration to determine the point of intersection between a field line from a point charge and a plane. This would involve setting the electric flux calculated using Gauss' law equal to the electric flux calculated using the electric field equation, and solving for the distance at which the field line becomes parallel to the plane. Then, using integration, we can determine the exact point of intersection.