Problem 11 - University Physics with Modern Physics

Question

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is 10.0 \(\mathrm{pC}\) . The inner cylinder has radius \(0.50 \mathrm{mm},\) the outer one has radius \(5.00 \mathrm{mm},\) and the length of each cylinder is 18.0 \(\mathrm{cm} .\) (a) What is the capacitance? (b) What applied potential difference is necessary to produce these charges on the cylinders?

Step-by-Step Solution

Answer

(a) Capacitance is 4.376 pF. (b) Potential difference is 2.285 V.

Step 1: Understand the Parameters

Given parameters include the radius of the inner cylinder \( r_1 = 0.50 \) mm = 0.0005 m, the radius of the outer cylinder \( r_2 = 5.00 \) mm = 0.005 m, the length of both cylinders \( L = 18.0 \) cm = 0.18 m, and the charge magnitude on each cylinder \( Q = 10.0 \) pC = 10.0 × 10^{-12} C.

Step 2: Formula for Capacitance of Coaxial Cylinders

The capacitance \( C \) of a pair of coaxial cylinders is calculated using the formula \[ C = \frac{2 \pi \varepsilon_0 L}{\ln(\frac{r_2}{r_1})} \], where \( \varepsilon_0 = 8.854 \times 10^{-12} \mathrm{F/m} \) is the permittivity of free space.

Step 3: Calculate ln(r2/r1)

Compute the natural logarithm part of the formula: \( \ln(\frac{r_2}{r_1}) = \ln(\frac{0.005}{0.0005}) = \ln(10) \approx 2.302 \).

Step 4: Calculate Capacitance C

Plug the values into the formula: \[ C = \frac{2 \pi \times 8.854 \times 10^{-12} \times 0.18}{2.302} \]. Compute to get \[ C \approx 4.376 \times 10^{-12} \text{F} \text{ (or 4.376 pF)} \].

Step 5: Potential Difference Using Capacitance

Using the formula \( V = \frac{Q}{C} \), where \( Q = 10.0 \times 10^{-12} \) C, calculate the potential difference: \[ V = \frac{10.0 \times 10^{-12}}{4.376 \times 10^{-12}} \].

Step 6: Calculate Potential Difference V

Compute \( V = \frac{10.0}{4.376} \approx 2.285 \text{ V} \).

Key Concepts

Coaxial Cylinders

Coaxial cylinders are an essential concept when dealing with the structure of a specific type of capacitor. Imagine two hollow, cylindrical tubes placed one inside the other, much like a telescope extending outwards. The crucial distinction here is that these cylinders have different radii. The inner cylinder is completely enclosed by the outer one. Each of these cylinders in the setup carries an equal but opposite charge.
The inner cylinder is negatively charged, while the outer one carries a positive charge. This charge setup is typical in capacitors to store energy by maintaining a potential difference between the conductors. Such a cylindrical configuration is highly efficient for certain applications due to its ability to offer high capacitance with relatively large surface areas while occupying a small volume.
In practical uses, this design helps in ensuring minimal losses of energy as well as a uniform electric field between the coaxial cylinders. By manipulating the sizes of these cylinders and using materials with varying dielectric properties, coaxial cylinder capacitors can be tailored for specific functions in electrical circuits.

Capacitance Calculation

The calculation of capacitance for a pair of coaxial cylinders involves understanding how this property is determined by their geometric setup and material properties. Capacitance, symbolized as\( C \), measures a system's ability to store charge per unit voltage.
For coaxial cylinders, the formula to determine capacitance is:\[ C = \frac{2 \pi \varepsilon_0 L}{\ln(\frac{r_2}{r_1})} \]Here, \( \varepsilon_0 \) is the permittivity of free space, \( L \) is the length of the cylinders, and \( r_1 \) and \( r_2 \) are the radii of the inner and outer cylinders, respectively. This formula helps in calculating \( C \) by accounting for both the physical dimensions of the cylinders and the material between them.
One of the critical parts of this formula is the natural logarithm of the ratio of the outer to the inner radius, \( \ln(\frac{r_2}{r_1}) \). This term shows how the spatial distance between the two cylindrical surfaces influences the system's capacitance. Once the values of the dimensions and permittivity are known, the capacitance can be easily calculated, providing valuable insights into how effectively a capacitor can store charge.

Potential Difference

The potential difference in a capacitor context refers to the voltage across the two conductors, or plates, of the capacitor. It is a vital part of understanding how capacitors function, as it is directly related to the energy storage capability of the capacitor.
For coaxial cylinders, to find the potential difference \( V \) between the inner and outer cylinders, you can use the formula:\[ V = \frac{Q}{C} \]Where \( Q \) is the charge on the cylinders and \( C \) is the capacitance calculated earlier. This equation derives from the basic relationship in a capacitor, where the potential difference is simply the charge divided by the capacitance.
A higher potential difference between the inner and outer cylinder indicates a stronger electric field and more energy stored within the capacitor. Understanding this concept is crucial, as it can help in designing circuits where specific voltage ratings must be adhered to in order to ensure desired performance and safety standards. By controlling the potential difference, you can adjust how the capacitor interacts within an electrical network.