Deriving Electric Charge: Line, Ring, Cylinder, Cone, Sphere & Shell

In summary, the conversation is about deriving the formula for electric charge for different cases without using Gauss's Law. The cases discussed are an infinite line of charge, a semi-infinite line of charge, a ring of charge, a semi-ring of charge or any arc of charge, a disk of charge, and an infinite sheet of charge. The remaining cases to derive are a cylinder of charge, a cone of charge, a sphere of charge, and a shell of charge. The basic "building block" for these cases is the ring of charge in different coordinate systems. The individual discussing the problem has all the necessary information and just needs to set up the integrals. They note that a sheet of charge can be viewed as a disk of charge with
  • #1
Chris W
27
0
Hi all. I need some help.
Without using Gauss's Law, I have to derive the formula for the electric charge for different cases.
-I already did few cases such as:
1. Infinite line of charge
2. Semi - infinite line of charge
3. Ring of charge
4. Semi - Ring of charge or any arc of charge
5. Disk of charge
6. Infinite Sheet of Charge

-I need to derive:
7. Cylinder of charge
8. Cone of charge
9. Sphere of charge
10. Shell of Charge

I know that the basic "building block" for cases 7-10 is the ring of charge which I already have. Basically what I need is the setup of the integral for cases 7-10.

Thank you

Chris W
 
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  • #2
It's just an exercise in different coordinates. A cylinder has constant r, varying z and phi/theta in cylindrical coordinate, a cone has varying z, r and phi/theta in cylindricalm a sphere has varying, r, phi, theta in spherical and a shell has constant r, varying phi, theta in spherical
 
  • #3
Thanks!

Yeah. Looks like I have all I need now I just have to set up the integrals.

Thanks! wow that was quick ... lol


please see what I have so far ... I hope all is good!

Thank you one more time

see attachment
 

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  • Semi- infinite line of charge .jpg
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  • ring of charge.jpg
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  • #5
this is the rest of what I have so far

sheet of charge can be viewed as the disk of charge and letting the R go to the infinify
 

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  • disk of charge page 2of2.jpg
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  • semi-ring or arc of charge.jpg
    semi-ring or arc of charge.jpg
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Related to Deriving Electric Charge: Line, Ring, Cylinder, Cone, Sphere & Shell

What is electric charge?

Electric charge is a fundamental physical property of matter that causes it to experience a force when placed in an electric field. It can be positive or negative and is measured in Coulombs (C).

What is the difference between line, ring, cylinder, cone, sphere, and shell charges?

Line, ring, cylinder, cone, sphere, and shell charges are all different geometrical shapes that can have an electric charge. The main difference between them is their distribution of charge. A line charge has a charge uniformly distributed along its length, while a ring charge has its charge uniformly distributed along a circle. A cylinder charge has its charge uniformly distributed along the surface of a cylinder, and a cone charge has its charge uniformly distributed along the curved surface of a cone. A sphere charge has its charge uniformly distributed throughout its volume, and a shell charge has its charge only on the surface.

How is the electric field calculated for these different charge configurations?

The electric field for a line charge is given by the equation E = λ/(2πε_0r), where λ is the charge per unit length, ε_0 is the permittivity of free space, and r is the distance from the line charge. For a ring charge, the electric field is given by E = Q/(4πε_0r^2), where Q is the total charge and r is the distance from the center of the ring. The electric field for a cylinder charge is given by E = Q/(2πε_0rL), where Q is the total charge, r is the distance from the center of the cylinder, and L is the length of the cylinder. For a cone charge, the electric field is given by E = Q/(2πε_0rL), where Q is the total charge, r is the distance from the tip of the cone, and L is the length of the cone. The electric field for a sphere charge is given by E = Q/(4πε_0r^2), where Q is the total charge and r is the distance from the center of the sphere. Lastly, the electric field for a shell charge is given by E = Q/(4πε_0r^2), where Q is the total charge and r is the distance from the center of the shell.

How do these different charge configurations affect the electric potential?

The electric potential for a line charge is given by V = λ/(2πε_0) * ln(r/r_0), where λ is the charge per unit length, ε_0 is the permittivity of free space, r is the distance from the line charge, and r_0 is a reference distance. For a ring charge, the electric potential is given by V = Q/(4πε_0r), where Q is the total charge and r is the distance from the center of the ring. The electric potential for a cylinder charge is given by V = Q/(2πε_0L) * ln(r/r_0), where Q is the total charge, L is the length of the cylinder, r is the distance from the center of the cylinder, and r_0 is a reference distance. For a cone charge, the electric potential is given by V = Q/(2πε_0L) * ln(r/r_0), where Q is the total charge, L is the length of the cone, r is the distance from the tip of the cone, and r_0 is a reference distance. The electric potential for a sphere charge is given by V = Q/(4πε_0r), where Q is the total charge and r is the distance from the center of the sphere. Lastly, the electric potential for a shell charge is given by V = Q/(4πε_0r), where Q is the total charge and r is the distance from the center of the shell.

How are these concepts applied in real-world situations?

The concepts of line, ring, cylinder, cone, sphere, and shell charges are applied in various real-world situations, such as in the design and function of electronic devices, power grids, and lightning rods. Understanding the behavior of electric charges in different geometrical shapes is essential in the study and application of electromagnetism, which has numerous practical applications in modern technology and everyday life.

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