Calculating E-Field Close to a Metal Plate from an Infinite Charged Rod

[joker_900]

Homework Statement

An infinite, thin, uniformly charged rod (charge density lamda) is situated parallel to a metal plate at a distance d above it. Calculate the E-field close to the surface of the plane as a function of perpendicular distance to the rod.

The Attempt at a Solution

I have worked out that the E-field due to the rod at a point a perpendicular distance x from the rod is lamda/(2piex) where e is epsilon0. However I don't know how to get the E-field due to the induction in the plate.

Also this question is in a section where all the others so far have been using method of images, but I didn't think I could do this as it doesn't specify that the plate is held at zero potential.

Please help!

 

[anathema]

Thank you for your post. The problem you have presented is an interesting one, and I would be happy to offer some guidance on how to approach it.

First, let's address the issue of the E-field due to the induction in the plate. Since the plate is a conductor, it will experience a redistribution of charges when the charged rod is placed above it. This will result in an induced electric field in the plate, which will contribute to the overall electric field at points close to the surface of the plate.

To calculate the induced electric field, we can use the method of images. We can imagine a second, imaginary charged rod placed on the other side of the plate, at the same distance d from the surface. This imaginary rod will have the same charge as the real rod, but with opposite sign. This is known as an image charge, and it helps us solve the problem by creating a simpler geometry.

Now, we can use the superposition principle to calculate the total electric field at points close to the surface of the plate. This will be the sum of the electric field due to the real rod and the electric field due to the image rod. By using the formula you have already derived for the E-field due to the real rod, and the formula for the E-field due to a point charge, you should be able to obtain an expression for the total electric field as a function of the perpendicular distance x.

As for the fact that the plate is not held at zero potential, this should not affect your calculation. The method of images is a mathematical tool that allows us to simplify the problem, and the result we obtain will still be valid even if the plate is not at zero potential.

I hope this helps in your solution. Please let me know if you need any further clarification or assistance. Best of luck with your problem!

 

[Moetasim]

I would approach this problem by using the principles of electrostatics and Gauss's law to calculate the electric field close to the metal plate. First, I would consider the metal plate as an equipotential surface, meaning the potential is constant at all points on the surface. This is because the plate is a conductor and any excess charge on the plate will distribute itself evenly to maintain a constant potential.

Next, I would use Gauss's law to calculate the electric field at a point close to the plate. This law states that the electric field is proportional to the charge enclosed by a Gaussian surface. In this case, the Gaussian surface would be a cylinder with one end on the metal plate and the other end close to the charged rod. The charge enclosed by this surface would be the charge density of the rod multiplied by the length of the cylinder.

Using this approach, I would be able to calculate the electric field due to both the charged rod and the induced charge on the metal plate. This would give a complete understanding of the electric field close to the metal plate as a function of perpendicular distance to the rod.

In terms of using the method of images, it may not be necessary in this case as the metal plate is not specified to be held at a specific potential. However, if desired, the method of images could still be used to simplify the problem and provide a more intuitive understanding of the electric field.