A co-axial cable with a central core of 2 mm diameter passes through the centre of a cylinder made of dielectric material diameter 5 mm. The dielectric cylinder is coated from outside with a conducting sheath. The outer conducting sheath is at ground potential. Calculate the capacitance of the

Question

A co-axial cable with a central core of 2 mm diameter passes through the centre of a cylinder made of dielectric material diameter 5 mm. The dielectric cylinder is coated from outside with a conducting sheath. The outer conducting sheath is at ground potential. Calculate the capacitance of the co-axial cable of 1 m length. The relative permeability of the dielectric material is 2.5.

Accepted Answer

The electric field at a point with a distance r from the axis of the cable is
[latex]E_{r}=\frac{Q}{ 2 π ε_{0}ε_{r}r}[/latex]
where Q is the positive charge per unit length of the inner wire and −Q is the charge per unit length of the outer wire.
The potential difference between radii a and b is
[latex]V=-\int_{a}^{b}\frac{Q}{2 π ε_{0}ε_{r}} \frac{dr}{r}=\frac{Q}{2 π ε_{0}ε_{r}} \log \frac{b}{a} [/latex]
Capacitance of a meter length, [latex]C=\frac{Q}{V}=\frac{2 π ε_{0}ε_{r}}{\log\left\lgroup\frac{b}{a}\right\rgroup }[/latex]
Given , 2a = 2 mm or a = 1 mm
2b = 6 mm or b = 3 mm
[latex]µ_{r} = 2.5 µ_{0} = 8.854 × 10^{−12}[/latex]
Substituting values,
[latex]C=\frac{2×3.14×8.854×10^{-12}×2.5}{2.303 \log_{10}\frac{3}{1}}[/latex] F
[latex]C=\frac{6.28×8.854×2.5}{2.303 \log_{10}3}[/latex] pico farad=[latex]\frac{60.3}{\log_{10}3}=\frac{60.3}{0.477}=126.4[/latex] PF.

Question 1.11: A co-axial cable with a central core of 2 mm diameter passes...

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