Question

Thermocoax is a type of coaxial cable used for high-frequency filtering in cryogenic quantum computing experiments. Its stainless steel shield has an inner diameter of $0.35\mathrm{mm},$ and its Nichrome conductor has a diameter of $0.17\mathrm{mm}$. Nichrome is used because its resistance doesn't change much in going from room temperature to near absolute zero. The insulating dielectric is magnesium oxide $(\mathrm{MgO}),$ which has a dielectric constant of $9.7.$ Calculate the capacitance per meter of Thermocoax.

Short answer

The capacitance per meter of Thermocoax is approximately $1.083\times {10}^{-10}F/m$.

Step-by-step solution

First, we need to convert the diameters $D$ and $d$ from millimeters to meters which can be done by dividing the values by $1000$. $D_{converted}={\displaystyle \frac{0.35}{1000}}=3.5\times {10}^{-4}m$ $d_{converted}={\displaystyle \frac{0.17}{1000}}=1.7\times {10}^{-4}m$ #Step 2: Calculate the capacitance per unit length C'#

Next, we can plug the converted values of $D_{converted}$, $d_{converted}$, ${ϵ}_0$, and ${ϵ}_r$ into the expression for capacitance per unit length. $C^{\prime }={\displaystyle \frac{2\pi (8.854\times {10}^{-12}F/m)(9.7)}{ln({\displaystyle \frac{3.5\times {10}^{-4}m}{1.7\times {10}^{-4}m}})}}$ #Step 3: Calculate the value of C'#

Using the above expression from Step 2, we can solve for the capacitance per unit length: $C^{\prime }={\displaystyle \frac{2\pi (8.854\times {10}^{-12}F/m)(9.7)}{ln(2.0588)}}\approx 1.083\times {10}^{-10}F/m$ #Step 4: Convert capacitance per unit length to capacitance per meter#

Finally, since the expression for $C^{\prime }$ above is already expressed in capacitance per meter, the answer is the calculated value of $C^{\prime }$. $C=1.083\times {10}^{-10}F/m$ The capacitance per meter of Thermocoax is approximately $1.083\times {10}^{-10}F/m$.

Key concepts

Coaxial Cable

A coaxial cable is a type of electrical cable consisting of a central conductor, an insulating layer, and an outer conducting shield. These components are arranged concentrically, giving the cable its 'coaxial' name. In the case of the Thermocoax presented in the exercise, the central conductor is made of Nichrome and the shield is stainless steel, with magnesium oxide (MgO) used as the insulator.

Coaxial cables are widely used in various types of data and signal transmission due to their exceptional shielding and ability to minimize electromagnetic interference. The geometry of the cable plays a crucial role in its electrical properties, including the characteristic impedance and the capacitance per unit length. The capacitance is a measure of the cable's ability to store charge and is especially important in high-frequency applications such as cryogenic quantum computing, where precise control over electrical signals is paramount.

Dielectric Constant

The dielectric constant, also known as the relative permittivity, is a dimensionless number that describes how much electric field is 'slowed down' inside a medium compared to a vacuum. It is a critical factor in the calculation of capacitance, particularly when dealing with materials like magnesium oxide (MgO) used as an insulator in coaxial cables.

In the provided exercise, the dielectric constant of magnesium oxide is 9.7. This value is quite high, indicating that MgO can hold a large amount of electric field per unit volume, which results in a higher capacitance for the cable. This property is leveraged in applications requiring stable capacitance over a range of temperatures, such as those encountered in cryogenic quantum computing. The dielectric constant directly influences the efficiency and performance of electrical components by affecting the energy stored between conductors.

Cryogenic Quantum Computing

Cryogenic quantum computing refers to quantum computers that operate at cryogenic temperatures, often just a fraction of a degree above absolute zero. At these extremely low temperatures, quantum bits (qubits) can maintain their quantum states for longer periods, which is essential for the reliability and accuracy of quantum computations.

The exercise highlights the use of specific materials, such as Nichrome for the conductor and magnesium oxide as the dielectric, which are chosen for their stable electrical properties at cryogenic temperatures. This stability ensures that the capacitance of the coaxial cables remains consistent during experiments, which is of utmost importance in maintaining the coherence of qubits in cryogenic quantum systems. Understanding the interplay between these materials and their electrical characteristics, including calculated capacitance, is part of the foundation for successfully developing and advancing quantum computing technology.