Question

A coaxial cable has conductor dimensions of \(a=1.0 \mathrm{~mm}\) and \(b=2.7 \mathrm{~mm}\). The inner conductor is supported by dielectric spacers \(\left(\epsilon_{r}=5\right)\) in the form of washers with a hole radius of \(1 \mathrm{~mm}\) and an outer radius of \(2.7 \mathrm{~mm}\), and with a thickness of \(3.0 \mathrm{~mm}\). The spacers are located every \(2 \mathrm{~cm}\) down the cable. ( \(a\) ) By what factor do the spacers increase the capacitance per unit length? \((b)\) If \(100 \mathrm{~V}\) is maintained across the cable, find \(\mathbf{E}\) at all points.

Short answer

Answer: The dielectric spacers increase the capacitance per unit length by a factor of 5. The electric field at all points can be expressed as a function of the distance (r) from the center of the inner conductor with the following formula: \(\mathbf{E}(r) = \frac{100}{r \ln(\frac{2.7}{1.0})}\).

Step-by-step solution

Compute the capacitance per unit length without the dielectric spacers

To find the capacitance per unit length \(C_{0}\), we can use the formula for a coaxial cable, given as: \(C_0 = \frac{2\pi \epsilon_0}{\ln(\frac{b}{a})}\), where \(\epsilon_0 = 8.85 \times 10^{-12} \mathrm{F/m}\) is the vacuum permittivity, \(a\) is the inner conductor radius, and \(b\) is the outer conductor radius. Substituting the values into the formula, we get : $$ C_0 = \frac{2\pi (8.85 \times 10^{-12})}{\ln(\frac{2.7}{1.0})} $$

Compute the capacitance per unit length with the dielectric spacers

Since the dielectric spacers have a dielectric constant \(\epsilon_r\), the capacitance per unit length with the dielectric spacers \(C\) can be found by multiplying \(C_0\) with \(\epsilon_r\): $$ C = \epsilon_r C_0 = 5 \times \frac{2\pi (8.85 \times 10^{-12})}{\ln(\frac{2.7}{1.0})} $$

Find the factor by which the spacers increase the capacitance per unit length

To find the factor by which the spacers increase the capacitance per unit length, we simply divide \(C\) by \(C_0\): $$ \mathrm{Factor} = \frac{C}{C_0} = \frac{5 \times \frac{2\pi (8.85 \times 10^{-12})}{\ln(\frac{2.7}{1.0})}}{\frac{2\pi (8.85 \times 10^{-12})}{\ln(\frac{2.7}{1.0})}} = 5 $$

Calculate the electric field \(\mathbf{E}\) at all points

For a coaxial cable with a potential difference of \(V = 100 \mathrm{V}\) between the inner and outer conductors, the electric field \(\mathbf{E}\) can be calculated using the formula \(E = \frac{V}{r \ln(\frac{b}{a})}\), where \(r\) is the distance from the center of the inner conductor. Since we are asked to find the electric field at all points, we can express \(\mathbf{E}\) as a function of \(r\): $$ \mathbf{E}(r) = \frac{100}{r \ln(\frac{2.7}{1.0})} $$ In conclusion, the dielectric spacers increase the capacitance per unit length by a factor of 5, and the electric field \(\mathbf{E}\) at all points can be expressed as a function of the distance \(r\) from the center of the inner conductor using the formula above.

Key concepts

Dielectric Spacers in Coaxial Cables

Dielectric spacers are integral components in the design of coaxial cables, which are commonly used in the transmission of electronic signals. These spacers, shaped like washers, are positioned at regular intervals within the cable to maintain the concentric alignment of the inner and outer conductors. The main function of dielectric spacers is to keep the conductors apart, preventing electrical contact while affecting the cable’s capacitance characteristics.

Typically, the spacers are made from materials with a specific dielectric constant, represented by the symbol \( \epsilon_r \). The dielectric constant measures the material's ability to store electrical energy. A higher dielectric constant in spacers results in greater capacitance per unit length of the cable. In the provided example, the dielectric constant of \( \epsilon_r = 5 \) means that the spacers can significantly increase the capacitance compared to air or vacuum, thus improving the cable's efficiency in storing and transmitting electrical energy.

Capacitance Per Unit Length Calculation

The capacitance per unit length of a coaxial cable is an important electrical characteristic. It indicates how much charge a cable can store per unit length at a given potential difference between its conductors. The basic formula to calculate the capacitance per unit length for a coaxial cable without any dielectric material present is given by \( C_0 = \frac{2\pi \epsilon_0}{\ln(\frac{b}{a})} \), where \( \epsilon_0 \) is the permittivity of free space, and \(a\) and \(b\) are the radii of the inner and outer conductors, respectively.

To factor in the presence of dielectric spacers, the capacitance with dielectric \(C\) is calculated by multiplying the capacitance without dielectric \(C_0\) by the dielectric constant \( \epsilon_r \). This adjustment accounts for the material's ability to polarize in response to the electric field, leading to greater charge storage capability. Hence, by introducing dielectric spacers, the capacitance per unit length increases, ultimately enhancing the cable's performance.

Electric Field in Coaxial Cables

In coaxial cables, the electric field \( \mathbf{E} \) is the force per unit charge exerted by the potential difference between the conductors. For the calculation of the electric field at any radius \( r \) within the cable, the following formula is used: \( \mathbf{E}(r) = \frac{V}{r \ln(\frac{b}{a})} \), where \(V\) is the potential difference, and \( r \) is the radial distance from the center of the inner conductor. The electric field is inversely proportional to the radius, signifying that it is stronger near the inner conductor and weakens as it approaches the outer conductor.

The existence of an electric field in coaxial cables is fundamental for the transmission of electrical signals. Its consistent calculation across different points inside the cable ensures that signal integrity is maintained throughout the length of the cable. Understanding the behavior of this electric field is crucial for optimizing cable design and troubleshooting potential issues that may arise during the cable's operation.