What is the Permittivity of a Spherical Capacitor as a Function of Radius?

[parsifal]

Homework Statement

A spherical capacitor (shell radiuses a and b, a<b) has the space between the shells filled with a dielectric, and the permittivity changes as a function of the radius so that the energy density stays constant (when radius R=a, then e_r=e₁). Find the permittivity as a function of R.

Homework Equations

$$E=\frac{Q}{4 \pi \epsilon R^2}$$

$$\epsilon = \epsilon _0 \epsilon _r$$

Energy density:
$$u=\frac{1}{2} \epsilon E^2= \frac{1}{2} \epsilon _0 \epsilon _r \frac{Q^2}{16 \pi ^2 \epsilon _0^2 \epsilon _r^2 R^4}$$

The Attempt at a Solution

I don't have a clue how this should be done. Now, u reduces to:
$$u=\frac{1}{2} \epsilon E^2= \frac{1}{2} \epsilon _0 \epsilon _r \frac{Q^2}{16 \pi ^2 \epsilon _0^2 \epsilon _r^2 R^4} = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r R^4}$$

So I guess that leaves me with only e_r to play with, in order to get rid of the R^4, which in turn is required to be taken out if u is needed to be constant. But as it was initally required that when R=a then e_r=e₁ , and I don't know how I could get rid of the R.

 

[anathema]

First, let's define some variables:
- R: radius of the spherical capacitor (a ≤ R ≤ b)
- a, b: radiuses of the inner and outer shells, respectively
- ε0: permittivity of free space
- εr(R): permittivity of the dielectric as a function of R
- E(R): electric field strength as a function of R
- Q: charge on the inner shell

Next, let's use the formula for energy density (u) to find the relationship between E and εr(R):
u = 1/2 ε E^2 = 1/2 ε0 εr(R) E^2 = constant

Therefore, we can say that:
E^2 ∝ 1/εr(R)

Now, let's use the formula for electric field strength (E) to find the relationship between Q and E:
E = Q / (4πεR^2)

Substituting this into our previous equation, we get:
Q^2 ∝ 1/εr(R)

Since we know that when R = a, εr(R) = ε1, we can write:
Q^2 ∝ 1/ε1

And when R = b, εr(R) = ε2 (where ε2 is the permittivity at the outer shell), so we can write:
Q^2 ∝ 1/ε2

Combining these two equations, we get:
Q^2 ∝ 1/ε1 = 1/ε2

Therefore, we can say that ε1 = ε2.

Now, let's use the formula for permittivity (ε) to find the relationship between εr(R) and R:
ε = ε0 εr(R)

Substituting this into our previous equation, we get:
Q^2 ∝ 1/εr(R) = 1/(ε0 ε)

And since we know that ε1 = ε2, we can say that:
εr(R) = ε1 = ε2 = 1/E0 = constant

Therefore, the permittivity as a function of R is simply a constant value, which is equal to ε1 = ε2.

In conclusion, the permittivity (ε) as a function of R is constant and equal to ε1 = ε2, which satisfies the requirement that the energy density remains constant.

 

[ogb p]

I would approach this problem by first understanding the concept of permittivity and how it relates to capacitance. Permittivity is a measure of how easily an electric field can pass through a material. In a capacitor, the permittivity of the dielectric material between the plates affects the capacitance of the capacitor.

Next, I would carefully analyze the given information and equations to see how they relate to the problem at hand. The given equation for energy density involves both the permittivity and the electric field, so I would look for a way to express the electric field in terms of the radius R.

Using the equation for electric field for a spherical capacitor, E=Q/4πεR^2, I can substitute this into the equation for energy density to get:

u=\frac{1}{2} \epsilon E^2=
\frac{1}{2} \epsilon _0 \epsilon _r \frac{Q^2}{16 \pi ^2 \epsilon _0^2 \epsilon _r^2 R^4} = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r R^4} = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r} \left( \frac{1}{R^4} \right)

Now, I can use the given information that the energy density should remain constant to solve for the permittivity as a function of R. This means that the term in parentheses must also remain constant. So, I can set it equal to a constant, say k, and solve for the permittivity:

\frac{1}{R^4} = k
\Rightarrow R^4 = \frac{1}{k}
\Rightarrow R = \frac{1}{\sqrt[4]{k}}

Substituting this into the equation for permittivity, I get:

\epsilon = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r k}

Now, using the given information that when R=a, then er=e1, I can solve for the constant k:

\epsilon = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r k} = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r a^4}
\