Potential difference between two cylinders

[aaaa202]

I need help with the attached exercise. Problem is not that I am not able to do it. You can find the field of an infinite cylinder with Gauss' law and then use that to find the potential difference.
My solutions notes however do it in a slightly different way, still arriving at the correct results. They use the expression for the potential of an infinite line:

V(r) = −1/(4$\pi$$\epsilon$₀)(2Q/L) ln(r/R₀)

And simply use that to calculate the potential difference. Can someone explain to me why this would yield the same result?

 

[tiny-tim]

hi aaaa202!

They use the expression for the potential of an infinite line:

V(r) = −1/(4$\pi$$\epsilon$₀)(2Q/L) ln(r/R₀)

And simply use that to calculate the potential difference. Can someone explain to me why this would yield the same result?

an infinite uniformly charged line and an infinite uniformly charged cylinder clearly both have cylindrically symmetric potentials (and fields)

so any cylinder larger than both will be an equipotential surface, and so from Gauss' law, if the enclosed charge-per-length is the same, so will be the fields (and the potentials)

 

[aaaa202]

Hmm okay. So is it because you mightaswell view the field of an infinite cylinder with charge Q as that of an infinite line placed along the center of the cylinder with the same charge?

 

[tiny-tim]

outside the cylinder, yes!

 

[paranormal]

Both methods, using Gauss' law and using the potential of an infinite line, are valid approaches to finding the potential difference between two cylinders. The reason they yield the same result is because they are essentially just two different ways of mathematically representing the same physical concept.

Gauss' law is a general law that relates the electric flux through a closed surface to the enclosed charge. When applied to the problem of an infinite cylinder, it allows us to calculate the electric field at any point outside the cylinder. From there, we can use the equation E = -∇V to find the potential difference between two points.

On the other hand, the potential of an infinite line is a specific mathematical expression that describes the potential at any point along an infinite line of charge. In the case of two cylinders, we can use this expression to calculate the potential at any point along the axis of the cylinders, and then use the difference between these potentials to find the potential difference.

In essence, both methods are just different mathematical representations of the same physical concept. So, it is not surprising that they yield the same result. However, it is always good to have multiple ways of approaching a problem, as it allows for a deeper understanding and can be useful in different situations.