Potential of coaxial with two dielectrics?

[lam58]

The question asks to find the potential of the ungrounded outer conducting sheath of a coaxial cable when the inner conductor has a voltage of $$\frac{220}{\sqrt{3}} kV$$.

Coaxial conditions:
i) Central copper conductor of diameter 4.8 cm.
ii) Insulating layer of XPLE of thickness 2.3 cm and relative permittivity k = 2.2.
iii) Conducting lead sheath of thickness 2.9 mm.
iv) Insulating HDPE layer of thickness 5 mm and k = 2.4.
v) Outer coating negligible thickness in contact with soil.

My attempt:

I found the XPLE to have a capacitance of 310.70x10^-12 F/m.

To find the potential I assumed the charge between both conductors would equal 0, thus the charge on the inner conductor = -'ve charge on sheath. Hence I can use Q = CV to find the charge:

$$310.73x10^{-12} * \frac{220}{\sqrt{3}} = 3.95x10^{-5} C$$

Then to find potential between the two I use:

$$v = \frac{-Q}{2\pi\epsilon_0\epsilon_r}.ln(b/a)$$

$$\Rightarrow v = \frac{-Q}{2\pi * 8.8x10^{-12} * 2.4}.ln(0.0789/0.0739) = -19.5 kV$$

Which implies the potential on the lead sheath = $$\frac{220}{\sqrt{3}} kV + -19.5kV = 107.5kV$$

Have I done this right?

 

[mark726]

Yes, your calculations and approach seem correct. You have correctly used the capacitance formula to find the charge on the inner conductor and then used the potential formula to find the potential on the outer conducting sheath. Your final answer of 107.5 kV seems reasonable based on the given parameters. However, it is always a good idea to double check your calculations and make sure you have used the correct units throughout.