Question

The capacitance of a spherical capacitor consisting of two concentric conducting spheres with radii \(r_{1}\) and \(r_{2}\) \(\left(r_{2}>r_{1}\right)\) is given by \(C=4 \pi \epsilon_{0} r_{1} r_{2} /\left(r_{2}-r_{1}\right) .\) Suppose that the space between the spheres, from \(r,\) up to a radius \(R\) \(\left(r_{1}

Short answer

Answer: The final expression for the capacitance with a dielectric layer of radius R is: $$C = \frac{4\pi \epsilon_{0} r_{1} r_{2}}{9r_{2} - r_{1} -(9r_{1} - r_{2})\frac{R-r_{1}}{R}}$$

Step-by-step solution

Capacitance of sphere with two dielectrics

We divide the space between the conducting spheres into two regions: the region between \(r_{1}\) and R filled with dielectric whose permittivity is \(10 \epsilon_{0}\), and the region from R to \(r_{2}\) filled with dielectric whose permittivity is \(\epsilon_{0}\). Let's label the capacitances for these regions as \(C_1\) for the region from \(r_{1}\) to R and \(C_2\) for region from R to \(r_{2}\). Since the dielectric fills the region between the spaces, the capacitances \(C_{1}\) and \(C_{2}\) are in series. Therefore, the total capacitance C is given by: $$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$$

Find the capacitance for each region

The capacitance of a spherical capacitor filled with a dielectric is given by the formula: \(C_{dielectric} = 4\pi \epsilon r_{1} r_{2} / (r_{2} - r_{1})\) For the region between \(r_{1}\) and R filled with the dielectric of permittivity \(10 \epsilon_{0}\), we have: $$C_1 = \frac{4 \pi (10 \epsilon_{0}) r_{1}R}{R - r_{1}}$$ For the region between R and \(r_{2}\) filled with the dielectric of permittivity \(\epsilon_{0}\), we have: $$C_2 = \frac{4 \pi \epsilon_{0} R r_{2}}{r_{2} - R}$$

Total Capacitance

Now, we substitute the expressions for \(C_{1}\) and \(C_{2}\) in the total capacitance equation: $$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$$ $$\frac{1}{C} = \frac{R - r_{1}}{4 \pi (10 \epsilon_{0}) r_{1}R} + \frac{r_{2} - R}{4 \pi \epsilon_{0} R r_{2}}$$ After solving for C, we get: $$C = \frac{4\pi \epsilon_{0} r_{1} r_{2}}{9r_{2} - r_{1} -(9r_{1} - r_{2})\frac{R-r_{1}}{R}}$$

Limit \(R = r_{1}\)

As \(R \rightarrow r_{1}\), the numerator remains the same, while the denominator becomes \(9r_{2} - r_{1} - (9r_{1} - r_{2}) = 8(r_{2} - r_{1})\). So the total capacitance simplifies to: $$C = \frac{4 \pi \epsilon_{0} r_{1} r_{2}}{8(r_{2} - r_{1})}$$ which is the capacitance of the original spherical capacitor without dielectric, confirming our limit case.

Limit \(R = r_{2}\)

As \(R \rightarrow r_{2}\), the numerator remains the same, while the denominator becomes \(9r_{2} - r_{1}\). So the total capacitance simplifies to: $$C = \frac{4 \pi \epsilon_{0} r_{1} r_{2}}{9r_{2} - r_{1}}$$ which corresponds to the capacitance of the spherical capacitor filled entirely with the dielectric of permittivity \(10 \epsilon_{0}\), confirming our limit case. Thus, the final expression for the capacitance with a dielectric layer of radius R is: $$C = \frac{4\pi \epsilon_{0} r_{1} r_{2}}{9r_{2} - r_{1} -(9r_{1} - r_{2})\frac{R-r_{1}}{R}}$$

Key concepts

Capacitance with Dielectric

When you have a spherical capacitor that consists of two concentric conducting spheres, things can get interesting as we introduce materials that can increase its capacitance. This concept revolves around a capacitor's ability to store electrical charge, which can be significantly influenced by introducing a dielectric material, a substance that increases capacitance by reducing the electric field for the same charge.

In the case of our spherical capacitor, a dielectric material with a permittivity of 10 times the permittivity of free space \(\epsilon_0\) fills part of the space between the spheres. This dielectric boosts the capacitance in that specific region.

• The formula for calculating capacitance with a dielectric: \C = 4\pi \epsilon r_{1}r_{2}/(r_{2}-r_{1})\.
• In our scenario, the presence of two dielectrics in separate regions results in different capacitances \(C_1\) and \(C_2\).

The overall effect is a higher capacitance for the part filled with the stronger dielectric, reflected by its separate calculation before combining the total capacitance.

Series Capacitance Formula

Understanding how capacitances combine is crucial, especially when they are connected in series. For two capacitors placed in series, the total capacitance \(C\) is less than any individual capacitance.

This is because the potential difference across the capacitors adds up, leading to the expression:

• The formula for capacitors in series: \(\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}\).
• This means you effectively have multiple capacitors working together, as the charge on each is the same but the voltage spreads over them.

In our spherical capacitor problem, the two regions filled with different dielectric materials behave as series capacitors.
The approach helps accommodate for the varied effects each dielectric has over its respective region. This series calculation helps solve the overall behavior of such capacitors with multiple dielectric layers.

Dielectric Material in Capacitors

Dielectrics play a fundamental role in enhancing the capability of capacitors. These materials are electrically insulating and, when placed in a capacitor, can dramatically affect its properties.

Here's what happens when a dielectric material is introduced within a capacitor:

• Dielectric materials increase the effective capacitance, allowing the capacitor to store more charge for the same voltage.
• They work by reducing the effective electric field within the capacitor, which decreases the energy needed to store the same amount of charge.
• The permittivity \(\epsilon\) of a dielectric is key, displaying how much influence the material will have compared to empty space (where \(\epsilon_0\) is the permittivity of free space).

In our example with the spherical capacitor, a portion of the space is filled with a dielectric that influences the capacitance considerably, as shown by the adjustment in the formula. The inclusion of such materials varies the capacitance and modifies how energy is stored, enhancing overall efficiency and capacity.