Electric Field above a Quarter Disk

[Squire1514]

• 
  1. Problem Statement.

Find the Electrostatic Field at a distance z above the xy-plane and along the z-axis due to a constant surface charge σ confined to the region a < √(x²+y²) < b, 0 < α < pi/2.
In other words, find the Electric Field a distance z above a quarter disk occupying the region between the positive x and y axes from r = a to r = b.

• 2. Known Equations

E = kq/r²
dq = σdA
dA = pi/2 * r^' *dr^'
r = √(r^'2+z²)

• 3. Attempt

The lack of symmetry is what loses me in this question. I know I need to set up tiny quarter rings and then integrate from A to B but I don't know what to do about the vector notation.

 

[Khashishi]

You can break it into x, y, z coordinates. r^2 = (x^2 + y^2 + z^2).
Write it as a double integral over dx and dy.

 

[jtbell]

Note to the Homework Helpers: This was originally posted in the Classical Physics forum. I moved it here instead of deleting it because the OP showed some thought and posted in a format that sort of resembles the homework forums template. Carry on!

 

[Stealth95]

dA = pi/2 * r^' *dr^'

Firstly, here I think you must write [itex]\displaystyle{dA=rdrd\varphi }[/itex] ([itex]\displaystyle{\varphi }[/itex] is the azimuth). Note that if you have two charges at the same distance from the origin but in different angles they don't create the same field at the point along the z-axis. The two fields have the same magnitude but different direction! So you can not neglect the angle at the integration as you integrate vector to find the field.

• 3. Attempt

The lack of symmetry is what loses me in this question. I know I need to set up tiny quarter rings and then integrate from A to B but I don't know what to do about the vector notation.

You can solve it in such a way, but you have to find the field of the ring at first. So I think it's easier to use double integral in cylindrical coordinates (that's from where the [itex]\displaystyle{dA }[/itex] I write above comes from). It's almost the same method with yours but you can find immediately the total field of the disk.

To take into account that [itex]\displaystyle{\vec{E}}[/itex] is a vector it may help you to write Coulomb's law like that:
[tex]\displaystyle{d\vec{E}=\frac{dq}{4\pi \varepsilon _0}\frac{\vec{z}-\vec{r}}{\left|\vec{z}-\vec{r} \right|^3}=\frac{dq}{4\pi \varepsilon _0}\frac{\vec{z}-\vec{r}}{(r^2+z^2)^{3/2}}}[/tex]
where [itex]\displaystyle{\vec{z}}[/itex] is the point where you want to find the field position vector and [itex]\displaystyle{\vec{r}}[/itex] is [itex]\displaystyle{dq}[/itex] position vector.

You can analyze these vectors in the Cartesian unit vectors and then use a double integral in cylindrical. Note that it's difficult to integrate with cylindrical unit vectors because they are not constant!

 

[paranormal]

4. Solution

As a scientist, it is important to approach problems systematically and logically. In this case, we can break down the problem into smaller, more manageable parts. First, let's define the coordinate system and variables involved. We have a quarter disk occupying the region between the positive x and y axes, with a surface charge σ confined to the region a < √(x2+y2) < b. We are interested in finding the electric field at a distance z above the xy-plane and along the z-axis.

To begin, we can use the known equations provided to us. The first equation, E = kq/r2, represents the electric field due to a point charge q at a distance r. In this case, we will need to integrate this equation over the entire quarter disk to find the total electric field at point P (z-axis, distance z above xy-plane).

Next, we can use the second equation, dq = σdA, to relate the surface charge density σ to the differential area element dA. This will allow us to express the total charge q in terms of the surface charge density and the area of the quarter disk.

Now, we can use the third equation, dA = pi/2 * r' *dr', to express the differential area element dA in terms of the variable r', which represents the distance from the origin of a small quarter ring. This will allow us to integrate over the entire quarter disk.

Finally, we need to take into account the lack of symmetry in this problem. We can do this by using vector notation and expressing the electric field as a vector. We can define the x and y components of the electric field as Ex and Ey, respectively. Then, we can use the Pythagorean theorem to find the magnitude of the electric field, E = √(Ex2 + Ey2). We can also use trigonometric functions to find the direction of the electric field, θ = tan-1(Ey/Ex).

By integrating over the entire quarter disk and taking into account the lack of symmetry, we can find the electric field at point P. This approach allows us to break down the problem into smaller, more manageable parts and use known equations to find the solution.