How Is Electrical Potential Derived from a Ring Charge?

[MissP.25_5]

Hi.
I have no idea how to do this. It's complicated because the length here is a variable. I posted the same question before but there was a little mistake so I am reposting this again.

The picture below shows the potential due to ring charge.
Please show the full steps of deriving the equation of electrical potential shown in the picture. I have no idea how to start.

NOTE:
The electric potential of the revolving symmetrical ring electric charge related to the axis z as depicted in the diagram 5.3. It is commonplace to use complete circle integral function in the charge simulation method. If the position (height) of ring electric charge is Z, the radius is R, and the charge density is λ, the electric potential of the point P will be as represented in the next equation.

In the equation, l is the distance between the part of the ring charge dθ and P.

 

[Simon Bridge]

Hi.
I have no idea how to do this. It's complicated because the length here is a variable. I posted the same question before but there was a little mistake so I am reposting this again.

The picture below shows the potential due to ring charge.
Please show the full steps of deriving the equation of electrical potential shown in the picture. I have no idea how to start.

I can give you a hint where to start, but I am not allowed to provide full steps like you ask.

The electric potential of the revolving symmetrical ring electric charge...

... the picture does not show a revolving system - please check.

... related to the axis z as depicted in the diagram 5.3. It is commonplace to use complete circle integral function in the charge simulation method. If the position (height) of ring electric charge is Z, the radius is R, and the charge density is λ, the electric potential of the point P will be as represented in the next equation.

In the equation, l is the distance between the part of the ring charge dθ and P.

So how would you normally go about finding the potential due to a distribution of charge?

You problem seem to be the "l" is not fixed?
That is not a problem if you express l in terms of r and θ

 

[MissP.25_5]

I can give you a hint where to start, but I am not allowed to provide full steps like you ask.

... the picture does not show a revolving system - please check.


So how would you normally go about finding the potential due to a distribution of charge?

You problem seem to be the "l" is not fixed?
That is not a problem if you express l in terms of r and θ

It's ok, I have solved this problem!

 

[Simon Bridge]

Excellent - perhaps you can help out someone in a similar fix by posting the resolution?

 

[rezeew]

Hello,

Thank you for your question. Deriving the equation for electrical potential due to a ring charge can be a bit complicated, but I will do my best to explain the steps involved.

First, we need to understand that the electric potential at a point is defined as the amount of work required to bring a unit positive charge from infinity to that point. In other words, it is the amount of energy needed to move a charge to a specific location.

In this case, we are dealing with a ring charge, which means that the charge is distributed along a ring at a certain distance from the axis z. We can use the concept of superposition to calculate the potential at a point P due to each small element of charge on the ring. This is represented by the integral function in the equation.

To start, we can use Coulomb's law to calculate the electric field at a point P due to a small element of charge (dθ) on the ring. This is represented by the term dE in the equation. We then multiply this by the distance between the element of charge and point P (l) to calculate the work done in bringing that charge to point P. This is the first part of the integral function.

Next, we need to consider all the small elements of charge on the ring and integrate over the entire ring to get the total work done in bringing a unit positive charge from infinity to point P. This is represented by the second part of the integral function, which goes from 0 to 2π (the full circle).

Finally, we divide by the charge density (λ) and the distance between the point P and the ring (Z) to get the total electric potential at point P. This is represented by the final division in the equation.

I hope this explanation helps you understand the steps involved in deriving the equation for electrical potential due to a ring charge. Please let me know if you have any further questions.