Magnitude of electric field between two concentric oppositely charged

[sa1988]

Homework Statement

Magnitude of electric field between two concentric oppositely charged spheres.

Homework Equations

The Attempt at a Solution

So I first went with the simple electric field equation E=V/d

So I have the potential difference V between the two sphere surfaces. I can't do Latex but hyperphysics has it anyway:

So I know V.
And surely, in E=V/d, the distance d is simply b-a.

So finally I sub my V and d into the equation and I'm left with:

E = Q/(4πεab)

Which looks reasonable enough.. But all sources on the Internet tell me that the answer is simply:

E=Q/4πεr²

The real confusing thing for me here is... What is 'r' ? What is it the radius from? The radius from the centre? Or from the surface of the inner sphere? Or something else..??

I fear my problem may be with the fact that I used E=V/d, which is possibly only valid for charged plates, rather than spheres? Either way, I'm still a little confused about what the 'r' actually means in the correct answer.

Thanks for any guidance :thumbs:

 

[sa1988]

Should 'r' be defined such that:

a < r < b

?

 

[BvU]

The expression above the $\Delta V$ formula you show is that selfsame E=Q/(4πεr²) !
Read up on Gauss's theorem (ibidem, another two lines higher up, or Gauss's law)

Your E=V/d is a special case for a constant $\vec E$ all over the place.
Otherwise (here for instance) $\vec E = -\vec\nabla V$.

And r is simply the distance to the center.

 

[sa1988]

Oh dear oh dear! So yeah I guess my main error was for the E=Vd part.

I've been taught about E = -∇V before, and yeah that makes perfect sense come to think of it.

Electric/Magnetic fields is certainly my weakest subject at the moment. Must study harder!

Thanks for showing me that :thumbs:

 

[paranormal]

I would like to clarify that the equation E=V/d is only applicable for charged plates, as you have correctly mentioned. In the case of concentric spheres, the electric field between them is not constant and it varies with distance from the center. This is because the electric field lines are radial and the distance from the center to a point on the field line changes as we move along the field line.

Now, coming to the correct equation, E=Q/(4πεr^2), where r is the distance from the center, is known as Coulomb's law. It is the fundamental law that describes the relationship between the magnitude of electric charge, Q, and the distance, r, between the charges. In this case, r is the distance from the center of the spheres, as the electric field is radial.

I hope this clarifies your confusion. It is always important to understand the underlying principles and equations before using them in calculations. Keep up the good work!