To determine the position along the positive $ z $-axis where the electric field due to a uniformly charged circular loop is at its maximum, we follow these steps:

Expression for Electric Field ($ E $):

The electric field $ E $ at a point along the $ z $-axis for a charged circular loop can be expressed as:

$ E = \frac{KQr}{(x^2 + R^2)^{3/2}} $

Here, $ K $ is the Coulomb's constant, $ Q $ is the total charge, $ r $ is a constant involving charge distribution, $ x $ is the distance along the $ z $-axis, and $ R $ is the radius of the loop.

Maximizing the Electric Field:

To find where $ E $ is maximum, we take the derivative of $ E $ with respect to $ x $ and set it to zero:

$ \frac{dE}{dx} = 0 $

Solve for $ x $:

Solving the equation from the derivative, we find:

$ x = \frac{R}{\sqrt{2}} $

Substitute Given Radius:

Given the radius $ R = a\sqrt{2} $, substituting into the expression for $ x $:

$ x = \frac{\sqrt{2}a}{\sqrt{2}} = a $

Thus, the position along the positive $ z $-axis where the electric field is maximum is at $ x = a $.