a)
i) Potential for r < a: ![V(r)=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V%28r%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
ii) Potential for a < r < b: ![V(r)=\frac{\lambda}{2\pi \epsilon_0} ln\frac{b}{r}](https://tex.z-dn.net/?f=V%28r%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%20ln%5Cfrac%7Bb%7D%7Br%7D)
iii) Potential for r > b: ![V(r)=0](https://tex.z-dn.net/?f=V%28r%29%3D0)
b) Potential difference between the two cylinders: ![V_{ab}=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V_%7Bab%7D%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
c) Electric field between the two cylinders: ![E=\frac{\lambda}{2\pi \epsilon_0} \frac{1}{r}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%5Cfrac%7B1%7D%7Br%7D)
Explanation:
a)
Here we want to calculate the potential for r < a.
Before calculating the potential, we have to keep in mind that the electric field outside an infinite wire or an infinite cylinder uniformly charged is
![E=\frac{\lambda}{2\pi \epsilon_0 r}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%20r%7D)
where
is the linear charge density
r is the distance from the wire/surface of the cylinder
By integration, we find an expression for the electric potential at a distance of r:
![V(r) =\int Edr = \frac{\lambda}{2\pi \epsilon_0} ln(r)](https://tex.z-dn.net/?f=V%28r%29%20%3D%5Cint%20Edr%20%3D%20%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28r%29)
Inside the cylinder, however, the electric field is zero, because the charge contained by the Gaussian surface is zero:
![E=0](https://tex.z-dn.net/?f=E%3D0)
So the potential where the electric field is zero is constant:
![V=const.](https://tex.z-dn.net/?f=V%3Dconst.)
iii) We start by evaluating the potential in the region r > b. Here, the net electric field is zero, because the Gaussian surface of radius r here contains a positive charge density
and an equal negative charge density
. Therefore, the net charge is zero, so the electric field is zero.
This means that the electric potential is constant, so we can write:
![\Delta V= V(r) - V(b) = 0\\\rightarrow V(r)=V(b)](https://tex.z-dn.net/?f=%5CDelta%20V%3D%20V%28r%29%20-%20V%28b%29%20%3D%200%5C%5C%5Crightarrow%20V%28r%29%3DV%28b%29)
However, we know that the potential at b is zero, so
![V(r)=V(b)=0](https://tex.z-dn.net/?f=V%28r%29%3DV%28b%29%3D0)
ii) The electric field in the region a < r < b instead it is given only by the positive charge
distributed over the surface of the inner cylinder of radius a, therefore it is
![E=\frac{\lambda}{2\pi r \epsilon_0}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20r%20%5Cepsilon_0%7D)
And so the potential in this region is given by:
(1)
i) Finally, the electric field in the region r < a is zero, because the charge contained in this region is zero (we are inside the surface of the inner cylinder of radius a):
E = 0
This means that the potential in this region remains constant, and it is equal to the potential at the surface of the inner cylinder, so calculated at r = a, which can be calculated by substituting r = a into expression (1):
![V(a)=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V%28a%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
And so, for r<a,
![V(r)=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V%28r%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
b)
Here we want to calculate the potential difference between the surface of the inner cylinder and the surface of the outer cylinder.
We have:
- Potential at the surface of the inner cylinder:
![V(a)=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V%28a%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
- Potential at the surface of the outer cylinder:
![V(b)=0](https://tex.z-dn.net/?f=V%28b%29%3D0)
Therefore, the potential difference is simply equal to
![V_{ab}=V(a)-V(b)=\frac{\lambda}{2\pi \epsilon_0} ln(\frac{b}{a})](https://tex.z-dn.net/?f=V_%7Bab%7D%3DV%28a%29-V%28b%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20ln%28%5Cfrac%7Bb%7D%7Ba%7D%29)
c)
Here we want to find the magnitude of the electric field between the two cylinders.
The expression for the electric potential between the cylinders is
![V(r)=\int\limits^b_r {Edr} = \frac{\lambda}{2\pi \epsilon_0} (ln(b)-ln(r))=\frac{\lambda}{2\pi \epsilon_0} ln\frac{b}{r}](https://tex.z-dn.net/?f=V%28r%29%3D%5Cint%5Climits%5Eb_r%20%7BEdr%7D%20%3D%20%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%20%28ln%28b%29-ln%28r%29%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%20ln%5Cfrac%7Bb%7D%7Br%7D)
The electric field is just the derivative of the electric potential:
![E=-\frac{dV}{dr}](https://tex.z-dn.net/?f=E%3D-%5Cfrac%7BdV%7D%7Bdr%7D)
so we can find it by integrating the expression for the electric potential. We find:
![E=-\frac{d}{dr}(\frac{\lambda}{2\pi \epsilon_0} (ln(b)-ln(r))=\frac{\lambda}{2\pi \epsilon_0} \frac{1}{r}](https://tex.z-dn.net/?f=E%3D-%5Cfrac%7Bd%7D%7Bdr%7D%28%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%28ln%28b%29-ln%28r%29%29%3D%5Cfrac%7B%5Clambda%7D%7B2%5Cpi%20%5Cepsilon_0%7D%20%5Cfrac%7B1%7D%7Br%7D)
So, this is the expression of the electric field between the two cylinders.
Learn more about electric fields:
brainly.com/question/8960054
brainly.com/question/4273177
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