Uniformly polarized disk on a conducting plane.

[jegues]

Homework Statement

A uniformly polarized dielectric disk surrounded by air is lying at a conducting plane, as shown in Fig. 2.36. The polarization vector in the disk is, $$\vec{P} = P\hat{k},$$ the disk radius is a, and the thickness d. Calculate the electric field intensity vector along disk axis normal to the conducting plane (z-axis).

Homework Equations

The Attempt at a Solution

See figure attached.

I can follow the solution up until the point where they add the two fields due to the 2 discs.

Which portion is E1 and which portion is E2?

Could someone please show all the steps they skipped?

Thanks again!

 

[anathema]

Thank you for your question. I understand that you are having difficulty understanding the solution provided for the electric field intensity vector along the disk axis. Let me try to break down the steps and clarify the solution for you.

Firstly, it is important to note that the electric field intensity vector is a vector quantity that describes the strength and direction of the electric field at a given point. In this case, we are interested in calculating the electric field intensity along the z-axis, which is normal to the conducting plane.

To calculate the electric field intensity, we need to take into account the contribution from both the disk and the air surrounding it. The electric field intensity due to a uniformly polarized dielectric disk can be calculated using the following equation:

E1 = P/2ε0

Where P is the polarization vector and ε0 is the permittivity of free space.

Now, let's consider the electric field intensity due to the air surrounding the disk. Since the air is a non-polarizable medium, the electric field intensity is simply given by:

E2 = σ/ε0

Where σ is the surface charge density on the conducting plane.

To calculate the total electric field intensity at a point along the z-axis, we need to sum the contributions from both the disk and the air. Therefore, the total electric field intensity at a point z along the z-axis is given by:

E = E1 + E2 = P/2ε0 + σ/ε0

Now, let's substitute the values given in the problem into this equation. Since the polarization vector in the disk is given by \vec{P} = P\hat{k}, we can rewrite it as P = P\hat{k} = Pcos(90°)\hat{i} + Psin(90°)\hat{j} = 0\hat{i} + P\hat{j}. This means that the polarization vector only has a component along the y-axis, which is normal to the disk surface.

Substituting this into the equation for E, we get:

E = (P\hat{j})/2ε0 + σ/ε0 = (P/2ε0)\hat{j} + (σ/ε0)\hat{j} = (P/2ε0 + σ/ε0)\hat{j}

Since we are only interested in the electric field intensity along the z-axis, we can ignore the component along the y-axis and