- #1

phys194

- 5

- 2

- Homework Statement
- Find the magnetic induction in the magneto-optic fluid

- Relevant Equations
- B = -grad(psi), grad^2 psi = 0

Consider the static field configuration shown in the image. There are three layers: 0 = vacuum, 1 = magneto-optic fluid and 2 = covering shell. Each of these layers have their own permittivity and permeability (ε_i,μ_i) (isotrope). A uniform electric field

My attempt was as follows:

As

I'm having trouble finding which boundary conditions need to be satisfied by ψ. No initial values of this potential have been given. I would assume that we need to look at the boundaries between each layer and since the normal component of the magnetic induction continuous. The main problem for me here is that I'm not sure if layer 2 does actually have a magnetic field (resulting from

Moreover, I'm not sure what to do with the limiting cases: as y and x approach infinity, does the potential have to disappear? For Y(y) this would mean that Y(y)=constant, because the magnetic field (and thus induction, because of the isotropy of the problem) is uniform and constant for y = infinity and y = - infinity.

Thank you for taking the time to help me out.

**H_0**= H_0/sqrt(2) * (**e_x**+**e_y**) is applied in vacuum (layer 0) using an electromagnet. This results in a static field**H**in the magneto-optic fluid core (layer 1). I need to find the magnetic induction in the magneto-optic fluid, assuming that the fluid container is infinite in the x- and z-direction.My attempt was as follows:

As

**J**=**0**(free current density) everywhere and each of the layers (or rather, regions) are simply connected, we can work with a scalar magnetic potential ψ and**B**= - grad (ψ). As the problem is z-independent, I would go for separation of variables (in Cartesian coordinates). So, ψ(**r**) = X(x)Y(y) and from Gauss' magnetic law follows that the magnetic potential needs to satisfy the Laplace equation. Therefore, X"Y + XY"=0, resulting in solutions of the form X(x)=Asin(kx)+Bcos(kx) and Y(y) = Ce^(-ky)+De^(ky) for some constant k (or the other way around, i.e. X = e^... and Y=sin+cos?). Now, we know the general solution for ψ, and to have the particular solution for this problem, we need to apply boundary conditions that ψ has to satisfy.I'm having trouble finding which boundary conditions need to be satisfied by ψ. No initial values of this potential have been given. I would assume that we need to look at the boundaries between each layer and since the normal component of the magnetic induction continuous. The main problem for me here is that I'm not sure if layer 2 does actually have a magnetic field (resulting from

**H_0**), as this is not drawn on the image?Moreover, I'm not sure what to do with the limiting cases: as y and x approach infinity, does the potential have to disappear? For Y(y) this would mean that Y(y)=constant, because the magnetic field (and thus induction, because of the isotropy of the problem) is uniform and constant for y = infinity and y = - infinity.

Thank you for taking the time to help me out.