1. Given data

• Radius of cylindrical wire:
  $R = 4\text{ mm} = 4 \times 10^{-3}\text{ m}$
• Current density:
  $J = 4 \times 10^6\ \text{A m}^{-2}$

We need the current through the outer annular portion of the wire, i.e. between radial distances $\dfrac{R}{2}$ and $R$.

2. Area of the outer annular portion

The required cross-sectional area is $A = \pi R^2 - \pi\left(\frac{R}{2}\right)^2$

$A = \pi R^2\left(1 - \frac{1}{4}\right) = \frac{3}{4}\pi R^2$

Now, $R^2 = (4 \times 10^{-3})^2 = 16 \times 10^{-6} = 1.6 \times 10^{-5}$

So, $A = \frac{3}{4}\pi (16 \times 10^{-6}) = 12\pi \times 10^{-6}\ \text{m}^2$

3. Current through this portion

Using $I = J A$

we get $I = (4 \times 10^6)(12\pi \times 10^{-6})$

$I = 48\pi\ \text{A}$

4. Required integer

The question asks for the blank in $\underline{\hspace{2cm}}\pi\ \text{A}$

Hence the required integer is $48$

5. Comparison with stored answer

Stored correct answer = $48$

Our derived answer = $48$

So they agree.