Introduction:
In this problem, we are given a long straight wire of radius a carrying a steady current I. The current is uniformly distributed across the cross-section of the wire. We need to find the ratio of the magnitude of the magnetic field at a distance a/2 from the wire to the magnitude of the magnetic field at a distance 2a from the wire.

Concept:
To solve this problem, we will use Ampere's Law. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.

Solution:

Step 1: Finding the magnetic field at a distance a/2:
To find the magnetic field at a distance a/2 from the wire, we will consider a circular loop of radius a/2 centered at the wire.

Step 1.1: Applying Ampere's Law:
According to Ampere's Law, the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.

∮B.dl = μ0Ienc

Here, B is the magnetic field, dl is an infinitesimal element of length along the loop, μ0 is the permeability of free space, Ienc is the current enclosed by the loop.

Step 1.2: Simplifying the integral:
The magnetic field is constant along the circumference of the loop. Therefore, the line integral simplifies to:

B ∮dl = μ0Ienc

Since the magnetic field is tangential to the loop, the line integral becomes:

B(2π(a/2)) = μ0Ienc

Simplifying further, we get:

Bπa = μ0Ienc

Step 1.3: Finding the current enclosed:
To find the current enclosed by the loop, we need to calculate the current passing through the cross-section of the wire enclosed by the loop.

The wire has a uniform current distribution across its cross-section. Therefore, the current passing through the cross-section is given by:

Ienc = (I/A) * Area_enc

Here, I is the total current, A is the total cross-sectional area of the wire, and Area_enc is the area of the cross-section enclosed by the loop.

The cross-sectional area of the wire is given by:

A = πa^2

The area enclosed by the loop is given by:

Area_enc = π(a/2)^2

Substituting the values in the equation for current enclosed, we get:

Ienc = (I/πa^2) * π(a/2)^2
= I/4

Step 1.4: Calculating the magnetic field:
Substituting the value of current enclosed in the equation for magnetic field, we get:

Bπa = μ0(I/4)

Simplifying further, we get:

B = (μ0I)/(4a)

Step 2: Finding the magnetic field at a distance 2a:
To find the magnetic field at a distance 2a from the wire, we will consider a circular loop of radius 2a centered at the wire.

Step 2.1: Applying