Gauss's Law - Point Charges

[nosracsan]

Homework Statement

A point charge Q is located on the axis of a disk of radius R at a distance b from the plane of the disk (Fig. below left). Show that if one fourth of the electric flux from the charge passes through the disk, then R = sqrt(3)b.

PLEASE SEE ATTACHED WORD DOCUMENT FOR DIAGRAMS!

 

[Doc Al]

Where's your solution? (They give you a big hint.)

 

[kerimek]

I would like to begin by stating that Gauss's Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed within that surface. It is a powerful tool that helps us understand the behavior of electric fields and their sources.

In this problem, we are given a point charge Q located on the axis of a disk of radius R at a distance b from the plane of the disk. The problem states that one fourth of the electric flux from the charge passes through the disk. This means that the electric flux through the disk, Φ, is equal to one-fourth of the total flux emitted by the point charge Q.

Using Gauss's Law, we can express the electric flux Φ as:

Φ = Q/ε₀

Where Q is the charge enclosed within the surface and ε₀ is the permittivity of free space. In this case, the charge enclosed is the point charge Q.

Now, we can calculate the total flux emitted by the point charge Q by considering a spherical surface centered at the point charge. The radius of this spherical surface is b, since the point charge is located at a distance b from the plane of the disk. Therefore, the total flux emitted by the point charge Q is given by:

Φ_total = Q/ε₀

Next, we can calculate the flux through the disk by considering a cylindrical surface centered at the point charge with a radius R and a height h, as shown in the diagram. The height h is equal to the distance between the disk and the point charge, which is b. Therefore, the flux through the disk is given by:

Φ_disk = Q/ε₀

Now, since we know that one-fourth of the total flux emitted by the point charge Q passes through the disk, we can equate these two expressions and solve for R:

Q/ε₀ = (1/4)Q/ε₀

This simplifies to:

R = sqrt(3)b

Therefore, we have shown that if one fourth of the electric flux from the charge passes through the disk, then the radius of the disk is equal to sqrt(3) times the distance between the disk and the point charge.

In conclusion, this problem demonstrates the usefulness of Gauss's Law in solving problems involving point charges and closed surfaces. By using this law, we were able to relate the electric flux through the disk to the charge enclosed within the surface and