A coiaxial cable consists of two thin coaxial cylinders electrically connected at one end, an inner cylindrical conducting tube of radius a carrying a steady current l which is screened by an outer cylindrical conducting sheath of radius b which provides a return path. There is no dielectric medium present.
Use Ampere's theorem to derive the total magnetic energy stored in the space between the conductors, show that the inductance of a length l of the cable is
`L=(mu_(0)l)/(2pi)ln((b)/(a))`
In this cable (a = 5 mm, b = 10 mm, l = 1000 m) is now employed in a (resistanceless) LC circuit containing a capacitance C = 1000 `muF`, determine the period of oscillations (neglect the capacitance of the cable itself).

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understanding the Configuration
We have a coaxial cable with an inner cylindrical conductor of radius \( a \) carrying a steady current \( I \) and an outer cylindrical sheath of radius \( b \). We need to find the magnetic field in the region between the two cylinders.

### Step 2: Applying Ampere's Law
According to Ampere's Law, the magnetic field \( B \) at a distance \( r \) from the center (where \( a < r < b \)) is given by:
\[
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