Question

A very long cylinder, of radius a, carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder (in terms of P). Show that the field outside the cylinder can be expressed (in cylindrical coordinates) in the form

Answer 1

Answer:     $Eout= (1/2\epsilon)(R/r)^2(2((P . r )r-P)$

Explanation:

The requires extensive derivation. The equation has been attached and answer is as follows.

Let the positively charged end as +d/2 and negatively charged end to be       -d/2.

Look for the Gauss's law in images attached which is E= (ρ/2ε)r

The formulas in the image below provides the formulas for the cylinders which are offset.

The total field inside the cylinder is now equals to adding the two is now  equals to             Ein= -P/2ε

In the second picture of the sketch attached it shows that the field outside is one of two lines of charge with linear densities of λ and offset by d:

Means that, λ=ρR² π

Continue along image no 2 and 3 attached to figure out the correct identities used in order to get to the required solution.

Remember the first images are the drawn sketches in order to imagine the electric field along the surface ( which here is the cylinder ).

Answer 2

Answer:

For electric field inside cylinder, check image 02 attached

For electric field outside cylinder, check image 03 attached

Explanation:

Let's consider the polarized cylinder as superposition of two cylinders with opposite,equal, uniform charge densities in a way shown in the figure in the "image 01"d attached ;

In general, if we have an object with polarization (P¬) , then we have to take two objects with similar shape to the system, with opposite, equal, and uniform charge densities and then we super-impose these two objects in such a way that the total dipole moment of this superimposed system is equal to the total dipole moment of original system.

Now, we can take the super- imposed system as equivalent to the original system for calculating electric field and potential.

Therefore,

For the electric field inside the cylinder, check the solution in "image 02" i attached

For the electric field outside the cylinder, check, "image 03" i attached.