a thin wire with linear charge density _ is surrounded by

a conducting cylindrical shell.
(a) If the electric field must be zero inside a conductor, is the electric field due
to the wire shielded from extending beyond the conducting shell?
(b) Find the electric field as a function of distance r from the thin wire.

To answer these questions, we'll apply Gauss's Law and the concept of electric field due to a line of charge.

(a) If the electric field must be zero inside a conductor, then any charge enclosed by the surface of a conductor should be zero. Therefore, the electric field due to the wire is shielded from extending beyond the conducting shell. Inside the conducting shell, the electric field is zero.

(b) To find the electric field as a function of distance 'r' from the thin wire, we need to consider a Gaussian surface in the form of a cylindrical surface coaxial with the wire. This surface will have a radius 'r' and a length 'L'.

Using Gauss's Law, we can write:

∮(E * dA) = (Q_enclosed) / ε₀,

where ∮ represents the closed surface integral, E is the electric field, dA is an infinitesimal area vector, Q_enclosed is the charge enclosed by the surface, and ε₀ is the electric constant.

Since the electric field is radial, its magnitude is constant over the curved surface of the Gaussian cylinder. Therefore, the flux through the curved surface is zero.

The only contribution to the flux will come from the two circular faces of the Gaussian cylinder:

E * 2πrL = (Q_enclosed) / ε₀.

Now, let's calculate the charge enclosed, Q_enclosed:

The linear charge density of the thin wire is represented by λ (lambda).

Q_enclosed = λ * L,

where L is the length of the Gaussian cylinder.

Plugging this back into the equation:

E * 2πrL = (λ * L) / ε₀.

Simplifying:

E = λ / (2πε₀r).

So, the electric field as a function of distance 'r' from the thin wire is given by:

E = λ / (2πε₀r).