Question

PHYNI2 Electromagnetism and Modern Physics Semester 2, 2019 Problem 2 square charged plate: rar, y, z). Ar. top view side view Useful integral: Va+ajve te tri de artan va where a, b, and care constants The figure shows a square plate of side-length L charged with a uniform surface charge density It is centred at r y =:= 0 and orientated in the plane. Your task is to determine the 3-component of the electric field at the point (..) =(,0,). offset along the -axis (at 0 and at a height d above the plane of the plate. Our strategy is to break the plate into a set of thin strips, with the strip having thickness ar and position along the axis (see figure). The electric fields of the strips can then be individually determined, and summed up via an integral to obtain the total electric field. In the y = 0 plane bisecting the plate, the electric field magnitude of strip i is = sino E LA, 21-1/2 or| is the distance from the centre of the strip to the point where Part A (6) Draw two sketches of the system in the y 0 plane (side view), the first showing the electric field lines of the plate, and the second showing the totel electric field vector at point as well as its s-component (b) Using your field vector sketch along with the expression for E above, show that in the plane the s-compnent of the electric field from strip i is DnLdAr, 21 (E). +7 0 plane (el Now determine the component of the total electrie bield of the square plate in the y You may find the integral given in the figure useful.

Answer 1

This is square charged plate whose dimension is $(x_{1}, y_{1}, z_{1})=(0,0,0)$ to $(x_{2}, y_{2}, z_{2})=(L,L,0)$ . In other words,let us mark this square charged plate as distinctly as possible:

(0,1,0) (L,L,0) (0,0,0) (L,0,0)

Here, the question has trick hidden in the question: This mentions a strip at the i^th position called x_i and it is merely the name (not an useful quantity during mathematical analysis). So we need to effectively deal with the term $\Delta x_{i}$ where $\forall$$\exists$$i\in [0,L]$, the [] indicating that the analysis is in closed space.

Given here that the surface charge density is $\eta$. This is not the usual symbol for charge density, though. But the catch here is that it stands by the definition: $\eta = \frac{Q}{A}$ .

Observe the triangle for the presence of r_i and d. This can be formulated into a Pythogorian case

Ο ΔΧ,

So we are essentially having an unknown across the main axis, the x-axis. We need to determine this in order to have the entire function representable for the sake of analysis

So:

$r_{i}^{2}=d^{2}+ x^{2}$

$x^{2}= r_{i}^{2}-d^{2}$

$x= \sqrt{r_{i}^{2}-d^{2}}$.

Let us have a clear picture of this complete sequence

Ο ΔΧ: