To determine the magnetic field at the center of a circular loop carrying current (I) and radius (R), we use the Biot-Savart Law and integrate over the entire loop.

Step 1: Biot-Savart Law Basics

The magnetic field (dB) due to a small current element (Id\vec{l}) at a distance (r) is:
[dB = \frac{\mu_0}{4\pi} \cdot \frac{I \, d\vec{l} \times \hat{r}}{r^2}]
For a circular loop at its center:

• (r = R) (distance from any element to the center),
• (d\vec{l}) (tangent to the loop) and (\hat{r}) (radial) are perpendicular ((\sin 90^\circ =1)),
• All (dB) vectors point in the same direction (perpendicular to the loop plane, via right-hand rule).

Step 2: Integrate Over the Loop

The magnitude of (dB) simplifies to:
[dB = \frac{\mu_0 I \, dl}{4\pi R^2}]

Integrate over the full circumference ((\int dl = 2\pi R)):
[B = \int dB = \frac{\mu_0 I}{4\pi R^2} \cdot 2\pi R = \frac{\mu_0 I}{2R}]

Final Result

The magnetic field at the center of the circular loop is:

[ \boxed{B = \frac{\mu_0 I}{2R}} ]

Direction: Perpendicular to the loop plane (outward if current is counterclockwise, inward if clockwise).

(\boxed{\dfrac{\mu_0 I}{2R}})

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