Electric potential constant within a conductor?

[Plamo]

1. Electric potential inside a conductor / outside a coaxial cable

Electric Potential inside a conductor(spherical) is a constant, although electric field is zero. How does that make sense given:
Given V=- \int E \cdot dl?
The integral should be 0. Even if you consider constants of integration, shouldn't they drop off because the integral is from the radius to 0?

Given that potential is non-zero inside a conductor, does the same hold true outside a coaxial cable? A Gaussian surface around the cable shows that the electric field outside the cable is 0. Do we have the same case where the potential is non-zero outside of the cable?

Homework Equations

V=- \int E \cdot dl

The Attempt at a Solution

The problem statement is my attempt at the solution. More of a lack of confusion than an actual problem.

Edit:
To clarify, this makes sense in reverse: E = del(V). Derivative of a constant is 0. How did that constant get there in the first place though?

 

[Matterwave]

That integral is certainly 0 within the conductor. With the limits of integration:

V_2-V_1=\int__1^2 \vec{E}dl=0

Obviously it's true since E is 0 inside the conductor...therefore the potential must be constant inside.

I guess I don't quite get what the problem is.

 

[Plamo]

The issue is that I don't see why it's non-zero inside. More importantly, I can't decide on whether or not it's 0 outside a coaxial cable.

 

[Matterwave]

What's non-0 inside? The electric field is definitely 0 inside a conductor (for electro-statics anyways).

How is your coaxial cable set up? Current moving in one direction inside and current moving in the opposite direction outside?

 

[jambaugh]

Consider two thick conducting plates connected to a battery so they have distinct constant potentials. If one is at potential zero the other is certainly not zero.

The potential equation you've given is more properly written:
\Delta V = V_2 - V_1 = -\int_{p_1}^{p_2} E\cdot dl
In short it defines a potential difference.