Calculate dielectric constants at an angle

[daskywalker]

Homework Statement

For an application you require a thin wafer of dielectric with dielectric constant = 12 perpendicular to the wafer. You have available a tetragonal crystal of dielectric ceramic with relative dielectric constants εx = εy = 18. and εz = 10. At what angle to the z axis will you cut the wafer? (this is the angle between the z-axis and the normal to the plane of the wafer).

Homework Equations

D=E_ij*ε_ij*ε_0 (in vector form)

The Attempt at a Solution

I tried to break it down into the components like I would do in a 2D case where D1=E1*ε11*ε0 (repeat for 2 and 3) where E1=Ecosθ and E2=sinθ. Now I do not know what E3 should be. I am sure it is not tanθ...Maybe I am wrong overall. If anyone can help me out that would be great.

 

[mark726]

Thank you for your question. To determine the angle at which the wafer should be cut, we need to consider the electric field components in each direction. In this case, we have a tetragonal crystal with relative dielectric constants of εx = εy = 18 and εz = 10. This means that the dielectric constant is different in each direction, and we cannot simply use the same equations as in a 2D case.

To solve this problem, we need to use the general form of the electric displacement vector, D=E*ε*ε0, where ε is the relative dielectric constant in the given direction. In this case, we have three components of the electric field, E1, E2, and E3, in the x, y, and z directions, respectively. The electric displacement vector in each direction is given by the following equations:

D1 = E1 * εx * ε0
D2 = E2 * εy * ε0
D3 = E3 * εz * ε0

Now, we know that the electric displacement vector is perpendicular to the wafer, which means that the components of D in the x and y directions must be equal (D1 = D2). This leads to the following equation:

E1 * εx * ε0 = E2 * εy * ε0

Since εx = εy = 18, we can simplify this to:

E1 = E2

This means that the electric field components in the x and y directions are equal. Now, to find the angle at which the wafer should be cut, we can use the following equation:

tanθ = E2/E3

Substituting E2 = E1 and εz = 10, we get:

tanθ = E1/10

Since we know that E1 = E2, we can rewrite this as:

tanθ = E1/E1

tanθ = 1

Therefore, the angle at which the wafer should be cut is 45 degrees with respect to the z-axis. This can also be confirmed by using the Pythagorean theorem to find the magnitude of the electric field in the x and y directions, which is equal to E1 and E2:

E1^2 + E2^2 = E3^2

E1^2 + E1^2 = E3^2

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