For which values of \(U\) is the magnitude of \(\Phi_{\text {cir }}\) independent of the radius \(r\) ?

Imagine standing in a vast, open field with a single strong magnet at the center. Similar to how a magnet creates a field around it, electric charges create an electric field in the space around them. A radial electric field is a special type of electric field that emanates outward from a point charge, just like the sun's rays spreading out in all directions.

The strength of a radial electric field emanating from a point charge decreases with distance from the charge, akin to how the warmth of the sun feels less intense the further away you are. Mathematically, we often express this field as a function of the distance, denoted by the variable r. In problems like the one under consideration, if the electric field's intensity U remains constant regardless of r, it implies that the influence or 'push' experienced by a charge within this field is the same no matter how far away it is from the source.

When problem-solving, this characteristic can significantly simplify calculations. If U(r) does not change with r, it means that the integral calculations over a path can be simplified as the variable field component does not need to be considered.

Picture yourself on a merry-go-round, holding tight to a horse as it circles around. Your path on the merry-go-round traces a circle, similar to the circular path we encounter in this problem. The direction you're facing at any point on that ride can be thought of as a unit tangent vector, designated as vec{t} in our calculations.

Mathematically, the unit tangent vector is crucial as it tells us the direction of a path at a particular point, offering directionality to the movement. In a circular path, this vector is always perpendicular to the radius and points in the direction of motion along the circle's edge. Because it's a 'unit' vector, its length is normalized to one, meaning it gives us pure direction without concern for how 'strong' or 'fast' the path is being traced.

Importance in Calculations

When computing integrals of vector fields along a path, the unit tangent vector aligns the field to the direction of the path, ensuring that only the component of the field that's parallel to the path is measured. This alignment is what gives the unit tangent vector its significance in line integrals.

Now combine your understanding of both a radial electric field and the unit tangent vector as we explore the concept of the line integral of a vector field. It's like adding up all the tiny pushes and pulls you feel as you walk through the electric field around a charged object, following a specific path.

The formal definition of a line integral may sound complex, but in essence, it's just a method to tally up these effects along a curve or path. For our circular path, the line integral calculates how the radial electric field's 'push' interacts with the circular motion we're considering, quantified over the entire loop. It’s like counting each step's contribution while walking along a path.

Finite vs. Infinitesimal Contributions

In practice, you divide the loop into infinitely small segments, taking each piece into account separately. ds represents such an infinitesimal piece of the path length, within which the force is assumed to be constant. As you might imagine, if the force (or our electric field in this example) is indeed constant across the entire path, the calculus simplifies greatly, and this is precisely the circumstance we explored in our exercise. In cases where the magnitude of U(r) is constant, the line integral doesn’t waver with radius changes - offering a simplified situation where the line integral’s value is solely influenced by the shape of the path and not by where it's situated with respect to the charged object.