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Cylindrical capacitors, interestingly, have their capacitance expressed in terms of length. This is because they are often quite long, and understanding capacitance in this way gives us a better idea of the capacitor's behavior over its length. The capacitance per unit length (denoted as \( C' \)) for a cylindrical capacitor is a measure of how well it can store an electric charge along each meter of its length.
To find \( C' \), we use the formula: \[ C' = \frac{2\pi\varepsilon_0}{\ln(b/a)} \]where:\

• \( \varepsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \) F/m.
• \( a \) is the radius of the inner conductor.
• \( b \) is the radius of the outer conductor.

In our exercise, the inner radius \( a \) is 2.2 mm and the outer radius \( b \) is 3.5 mm. These need to be converted to meters when plugged into the formula.
By calculating, you'll see that \( C' \approx 3.62 \times 10^{-11} \) F/m, which tells us that for each meter of length, the capacitor can hold \( 3.62 \times 10^{-11} \) Farads of charge. This is an essential characteristic used in design and application.

The potential difference in a cylindrical capacitor is the voltage between the inner and outer conductors. It's a crucial aspect as it directly influences how much charge the capacitor can hold. This potential difference is measured in volts and, in our scenario, the inner conductor is 350 mV, or 0.35 V, more positive than the outer conductor.
We utilize this voltage difference to further calculate the charge distribution. By the relation \( V = q'/C' \), where \( V \) is the potential difference, \( q' \) is the charge per unit length, and \( C' \) is the capacitance per unit length, we can rearrange to determine \( q' \):\[ q' = C' \cdot V \]Using \( C' \approx 3.62 \times 10^{-11} \) F/m from our previous calculation and the given \( V = 0.35 \) V, we find \( q' \approx 1.27 \times 10^{-11} \) C/m. This indicates how much charge is distributed along each meter of the capacitor given the existing voltage difference.

Charge distribution in a cylindrical capacitor describes how electric charge is positioned along its length. The formula \( Q = q' \cdot L \), where \( Q \) is the total charge, \( q' \) is charge per unit length, and \( L \) is the total length of the capacitor, allows us to calculate the charge managed by the entire capacitor.
In the given problem, the capacitor is 2.8 meters long. Using the previously calculated \( q' \approx 1.27 \times 10^{-11} \) C/m, we find the total charge \( Q \approx 3.56 \times 10^{-11} \) C.
The charge distribution is symmetric. It means the inner conductor has a positive charge of \( +3.56 \times 10^{-11} \) C, while the outer conductor has an equal and opposite charge, \( -3.56 \times 10^{-11} \) C. This negative charge on the outer conductor balances the positive charge, ensuring the entire capacitor is electrically neutral. This understanding of charge distribution is vital for effectively designing and using capacitors in electronic circuits.