Question

A spherical capacitor is made from two thin concentric conducting shells. The inner shell has radius \(r_{1}\), and the outer shell has radius \(r_{2}\). What is the fractional difference in the capacitances of this spherical capacitor and a parallel plate capacitor made from plates that have the same area as the inner sphere and the same separation \(d=r_{2}-r_{1}\) between plates?

Short answer

Answer: The fractional difference in the capacitances of the spherical capacitor and the parallel plate capacitor is given by \(\frac{r_2 - r_1}{r_1}\), where \(r_1\) and \(r_2\) are the radii of the inner and outer spheres of the spherical capacitor, respectively.

Step-by-step solution

Capacitance of a Spherical Capacitor

To calculate the capacitance of a spherical capacitor made from two thin concentric conducting shells with radii \(r_{1}\) and \(r_{2}\), we use the following formula: \(C_\text{spherical} = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1}\) where \(\varepsilon_0\) is the vacuum permittivity.

Capacitance of a Parallel Plate Capacitor

To calculate the capacitance of a parallel plate capacitor with the same area as the inner sphere and the same separation \(d = r_2 - r_1\) between the plates, we use the following formula: \(C_\text{parallel} = \frac{\varepsilon_0 A}{d}\) where \(A\) is the area of the plates which is equal to the surface area of the inner sphere: \(A = 4 \pi r_1^2\) Therefore, the capacitance of the parallel plate capacitor is: \(C_\text{parallel} = \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}\)

Find the Difference in Capacitances

We will now find the difference between the capacitances of the spherical and parallel plate capacitors: \(\Delta C = C_\text{spherical} - C_\text{parallel}\) Substituting the values, we get: \(\Delta C = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} - \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}\)

Calculate the Fractional Difference

Finally, we will calculate the fractional difference between the capacitances. The fractional difference is given by: \(\text{Fractional Difference} = \frac{\Delta C}{C_\text{parallel}}\) Substituting the values, we get: \(\text{Fractional Difference} = \frac{4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} - \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}}{\frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}}\) Simplifying, we get: \(\text{Fractional Difference} = \frac{r_2 - r_1}{r_1}\) This is the fractional difference in the capacitances of the spherical capacitor and the parallel plate capacitor.

Key concepts

Spherical Capacitor

A spherical capacitor consists of two concentric spherical conducting shells. These shells have different radii, with the inner shell sized at radius \( r_1 \) and the outer shell at radius \( r_2 \). The unique geometry of a spherical capacitor allows it to store electric charge between the two shells. Since the shells are concentric, the electric field is radial, and this influences how we calculate its capacitance.
To find the capacitance of a spherical capacitor, we utilize the formula:

• \( C_\text{spherical} = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} \)

Here, \( \varepsilon_0 \) is the vacuum permittivity, a constant that quantifies the ability of the vacuum to permit electric field lines. This formula illustrates how the capacitance depends significantly on the product and difference of the radii of the inner and outer shells. Moreover, the larger the inner radius or the smaller the gap between shells, the greater the capacitance becomes.

Parallel Plate Capacitor

A parallel plate capacitor differs from a spherical capacitor by its geometry. It typically comprises two large, flat conducting plates positioned parallel to one another with a separation \( d \). In our specific problem, the parallel plate capacitor is set up to share the same surface area as the inner sphere and the same gap \( d = r_2 - r_1 \) as the spherical capacitor.
The capacitance of a parallel plate capacitor can be calculated by the formula:

• \( C_\text{parallel} = \frac{\varepsilon_0 A}{d} \)

Here, \( A \) represents the area of the plates. For this case, the area corresponds to the surface area of the inner sphere, \( A = 4 \pi r_1^2 \). Therefore, the special relationship between plate area and separation allows the configuration to store charge easily if \( r_1 \) is large and \( d \) is small.

Fractional Difference

The concept of fractional difference is significant when comparing two similar entities but of different configurations, like the spherical and parallel plate capacitors. It is a measure detailing how much the capacitance of one system deviates as a fraction of the other. For our capacitors, the fractional difference is defined as:

• Fractional Difference \( = \frac{C_\text{spherical} - C_\text{parallel}}{C_\text{parallel}} \)

By plugging in the expressions for \( C_\text{spherical} \) and \( C_\text{parallel} \), and simplifying, the fractional difference formula becomes:

• \( \frac{r_2 - r_1}{r_1} \)

This result indicates how the capacitance of the spherical capacitor varies relative to the parallel plate capacitor depending on the dimensions of the radii \( r_1 \) and \( r_2 \). It highlights when one might be more significantly affected by changes in geometric configuration, pointing towards efficiency in different applications.