A coaxial transmission line has a solid inner conductor of radius a and a thin-walled outer conductor of radius b. The medium between the conductors has permeability (Mu)0. Find the inductance per meter of length of the line (a) if all current of the inner conductor is on its surface (at radius a) (actually a thin layer of negligible thickness) and (b) if the current of the inner conductor is uniformly distributed throughout its cross-sectional area, that is if the internal inductance of the inner conductor is taken into account.

To find the inductance per meter of length of a coaxial transmission line, we need to calculate the magnetic field inside the transmission line and then use it to determine the inductance.

(a) If all the current of the inner conductor is on its surface, the magnetic field outside the inner conductor is zero. Therefore, we only need to consider the magnetic field inside the inner conductor.

To calculate the magnetic field inside a conductor with a surface current, we will use Ampere's Circuital Law:

∮B·dl = μ₀I_enc

Where:
- B is the magnetic field
- I_enc is the enclosed current
- dl is a small differential length along the closed path
- μ₀ is the permeability of free space (μ₀ = 4π×10⁻⁷ Tm/A)

Since the current is distributed on the surface, we consider a circular path with radius r within the inner conductor. The enclosed current is the current on the surface of the inner conductor, which is I = 2πra, given that the current is uniformly distributed over the surface. Therefore, the equation becomes:

B(r) × (2πr) = μ₀ × 2πra

The magnetic field B(r) will be constant inside the conductor, so we can rewrite the equation as:

B × 2πr = μ₀ × 2πra

Simplifying and solving for B:

B = (μ₀a) / r

Now, to find the inductance per meter of length (L), we need to calculate the flux (Φ) threading the cross-section of the inner conductor and divide it by the current (I). The flux can be calculated as:

Φ = B × A

Where:
- A is the cross-sectional area of the inner conductor (A = πa²)

Substituting the values in, we get:

Φ = (μ₀a) / r × πa² = πμ₀a³ / r

Finally, we divide the flux by the current to obtain the inductance per meter:

L = Φ / I = (πμ₀a³ / r) / (2πra) = (μ₀a²) / (2r)

Therefore, the inductance per meter (L) of a coaxial transmission line with all current of the inner conductor on its surface is given by:

L = (μ₀a²) / (2r)

(b) Now, let's consider the scenario where the current of the inner conductor is uniformly distributed throughout its cross-sectional area, taking into account the internal inductance.

The internal inductance refers to the self-inductance of the inner conductor itself. To calculate the inductance per meter, we need to consider the magnetic field contributions from both the inner and outer conductors.

The magnetic field inside the inner conductor follows the same equation as in part (a):

B_inner = (μ₀a) / r

The magnetic field inside the outer conductor can be calculated using a similar approach. Considering a circular path between the inner and outer conductors with radius r₁, the enclosed current is I = 2πra²/b², given that the current is uniformly distributed over the cross-sectional area. Therefore, applying Ampere's Law, we get:

B_outer × (2πr₁) = μ₀ × (2πra²/b²)

Simplifying and solving for B_outer:

B_outer = (μ₀a²/b²) × r₁

To determine r₁, we consider that the total radius of the cross-section is b. Since the inner conductor has a radius of a, the outer conductor's radius r₁ is given by:

r₁ = b - a

Substituting this into the equation for B_outer:

B_outer = (μ₀a²/b²) × (b - a)

The total magnetic field inside the coaxial transmission line is the sum of the inner and outer fields:

B_total = B_inner + B_outer = (μ₀a) / r + (μ₀a²/b²) × (b - a)

Now, we can proceed to calculate the inductance per meter (L) by calculating the flux (Φ) and dividing it by the current (I).

The flux threading the cross-section of the inner conductor is given by:

Φ_inner = B_inner × A_inner = (μ₀a/r) × πa² = πμ₀a³ / r

The flux threading the cross-section of the outer conductor is given by:

Φ_outer = B_outer × A_outer = [(μ₀a²/b²) × (b - a)] × π(b² - a²)

Finally, we calculate the total flux and the inductance per meter:

Φ = Φ_inner + Φ_outer

L = Φ / I

Substituting the values and simplifying, we obtain the expression for the inductance per meter considering the internal inductance:

L = [(μ₀a³ / r) + (μ₀a²/b²) × π(b² - a²)] / [2πra]

Therefore, the inductance per meter (L) of a coaxial transmission line with the current uniformly distributed throughout the inner conductor's cross-sectional area, taking into account the internal inductance, is given by:

L = [(μ₀a³ / r) + (μ₀a²/b²) × π(b² - a²)] / [2πra]