# Solve Polarized Cylinder Griffiths Problem 4.13 Page 173

• stunner5000pt
In summary, the "Polarized Cylinder Griffiths Problem" is a problem from the textbook "Introduction to Electrodynamics" by David J. Griffiths, which involves finding the electric field and polarization charge density of a polarized cylinder. To solve this problem, one can use the method of separation of variables and apply boundary conditions, as well as have a good understanding of electric potential, electric field, polarization, and related concepts. This problem has real-life applications in electrical engineering, and there are helpful resources such as online forums, textbooks, and consulting with experts, to aid in solving it.
stunner5000pt
Griffith's problem 4.13 page 173

A very long cylinder of radius a carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Show taht the field outside the cylinder can be expressed in the form

$$\vec{E}(\vec{r}) = \frac{a^2}{2 \epsilon_{0} s^2} [2(\vec{P}\bullet\hat{s})\hat{s} - \vec{P}]$$

I m wondering if this is in any way similar to that of a sphere in that we can find the potnetial (from the solution of Laplace's equation) and from there we can find the electric field both inside and outside.

Is there a shorter method, though?

problem 4.14
When you polarize a neutral dielectric, charge moves a bit, but the total remains zero. Thi fact should be reflected i nteh bound charges simga b and rho b. Prove from the given equations taht the total bound charge vanishes
$$\sigma_{b} = \vec{P} \bullet \hat{n}$$
$$\rho_{b} = - \vec{\nabla} \bullet \vec{P}$$

wel ok
$$\sigma_{b} = (P_{x},P_{y},P_{z}) \bullet (n_{x},n_{y},n_{z})$$
and we know that
$$\frac{\partial P}{\partial n} = \vec{P}\bullet\hat{n}$$
but the thing is does this hold??

$$\frac{\partial P}{\partial n_{x}} = \frac{\partial P}{\partial x}$$??

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The short answer is yes, the method for finding the electric field inside and outside a cylinder with uniform polarization is similar to that of a sphere. Both involve solving Laplace's equation and using the boundary conditions to find the electric field. However, there is a shorter method for finding the electric field for a cylinder.

To find the electric field inside the cylinder, we can use the method of images. We can imagine a system consisting of two cylinders, one with uniform polarization and the other with opposite polarization. This results in a cancellation of the electric field inside the cylinder, leaving only the electric field outside.

To find the electric field outside the cylinder, we can use the method of images again, but this time with a plane of uniform polarization instead of a cylinder. This results in an electric field that can be expressed in the form given in the problem.

As for problem 4.14, we can prove that the total bound charge vanishes by using Gauss's law. We know that the divergence of the polarization vector is equal to the negative of the volume charge density. Therefore, if we integrate the divergence of the polarization over a closed surface, we will get the total bound charge enclosed by the surface. However, since the polarization vector is divergenceless, the integral will be equal to zero, proving that the total bound charge vanishes.

## 1. What is the "Polarized Cylinder Griffiths Problem"?

The "Polarized Cylinder Griffiths Problem" refers to problem 4.13 on page 173 of the textbook "Introduction to Electrodynamics" by David J. Griffiths. This problem involves finding the electric field inside and outside of a polarized cylinder, and determining the polarization charge density on the surface of the cylinder.

## 2. How do you solve the "Polarized Cylinder Griffiths Problem"?

To solve the "Polarized Cylinder Griffiths Problem", you can use the method of separation of variables to find the electric potential inside and outside of the cylinder. Then, you can apply boundary conditions to determine the polarization charge density on the surface of the cylinder. Finally, you can use the relation between electric field and potential to find the electric field inside and outside of the cylinder.

## 3. What are the key concepts involved in solving the "Polarized Cylinder Griffiths Problem"?

The key concepts involved in solving the "Polarized Cylinder Griffiths Problem" include electric potential, electric field, polarization, and boundary conditions. It is also important to have a good understanding of the method of separation of variables and the relation between electric field and potential.

## 4. Are there any real-life applications of the "Polarized Cylinder Griffiths Problem"?

Yes, there are real-life applications of the "Polarized Cylinder Griffiths Problem" in the field of electrical engineering. This problem can help in understanding the behavior of electric fields and potential in cylindrical systems, which can be useful in designing and analyzing electrical circuits and devices.

## 5. Are there any helpful resources for solving the "Polarized Cylinder Griffiths Problem"?

Yes, there are many helpful resources for solving the "Polarized Cylinder Griffiths Problem". Some suggestions include online forums, textbooks, and lecture notes on electromagnetism and electrostatics. It can also be helpful to consult with a professor or a fellow scientist who has experience in solving similar problems.

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