Finding Electric Field due to hemispherical charge distribution

[S[e^x]=f(u)^n]

Homework Statement

I need to find the electric field along the Z-axis for a hemispherical open shell of radius R (open face down). the shell is oriented so its symmetrical about the Z axis.

The Attempt at a Solution

What I've done is set it up, taken the vector from the origin (at the center of what would be the full sphere), and subtracted the vector bound to the z axis, to get the vector from the differential surface of the shell to an arbitrary point on the z axis. I then integrate, using spherical coordinates the 1/r^2 expression...

I can't seem to get this to work, can someone guide me through the right steps?, I've been trying to do this for about 7hrs now, and feel really stupid since I'm a 3rd year uni student majoring in physics...

 

[cupid.callin]

consider any thin circular strip on hemisphere at angle θ from vertical and subtending angle dθ on center.

find field due to this just like you do for ring on axis
then integrate it from θ = 0 to 90 degree

 

[S[e^x]=f(u)^n]

I've tried that, and i end up with E= [(surface charge density)*(R)^2]/[2*epsilon*(R-z)^2]

where R is the hemispheres radius, and z, is a point on the z axis...

i then integrate this to find the potential between R (along the z axis), and the origin, which ends up in a divide by 0 error... so i know the Electric field equation cannot be right although it seems to check out dimensionally...

is there an easier way to find the potential between the origin and R(on the z axis) that I'm missing? right now I'm just going after the electric field, which should then lead to a simple integration to get the potential difference between the two points. or have i forgotten something?

 

[gneill]

I've tried that, and i end up with E= [(surface charge density)*(R)^2]/[2*epsilon*(R-z)^2]

where R is the hemispheres radius, and z, is a point on the z axis...

Perhaps you could show how you arrived at that? The well-known result for the on-axis field produced by a ring of charge Q of radius a and distance x is given by:

$$E_x = \frac{{kQx}}{{\left( {x^2 + a^2 } \right)^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$

 

[rwisz]

I understand that sometimes solving a complex problem can be challenging and time-consuming. It is important to approach the problem systematically and consider all relevant factors. In this case, finding the electric field due to a hemispherical charge distribution may require using advanced mathematical techniques and considering the symmetrical nature of the shell.

One possible approach could be to use Gauss's law, which states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. In this case, the closed surface could be a hemisphere with its flat side facing down. By considering the symmetrical nature of the charge distribution, we can simplify the calculation and find the electric field along the z-axis.

Another approach could be to use the concept of superposition, where we can break down the hemispherical shell into smaller, simpler charge distributions and then use the principle of superposition to find the overall electric field. This method may require some additional calculations and consideration of the geometry of the shell, but it can be a useful approach in solving complex problems.

It is also important to double-check your calculations and make sure you are using the correct units and equations. Sometimes, a small mistake can lead to incorrect results, so it is always a good idea to review your work and seek help from a professor or peer if needed.

Overall, finding the electric field due to a hemispherical charge distribution can be a challenging but rewarding task. By approaching the problem systematically and considering all relevant factors, you can overcome any difficulties and successfully solve the problem.