State Biot-Savart Law. Derive an expression for this intensity of the magnetic field at the center of a current-carrying circular loop on its basis.

Biot-Savart Law
The Biot-Savart Law describes the magnetic field generated by a current-carrying conductor. It provides a relationship between the magnetic field and the current flowing through the conductor.
The law states that the differential magnetic field dBd\mathbf{B} at a point in space due to a small segment of current II is directly proportional to the current, the length of the segment, and the sine of the angle between the segment and the position vector to the point, and inversely proportional to the square of the distance between the current element and the point where the field is being calculated.
Mathematically, the Biot-Savart Law is expressed as:
dB=μ04πI dl×r^r2d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{r}}{r^2}
Where:
• dBd\mathbf{B} is the differential magnetic field at a point due to a small current element,
• μ0\mu_0 is the permeability of free space (μ0=4π×10−7 T\cdotpm/A\mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A}),
• II is the current passing through the conductor,
• dld\mathbf{l} is the differential length vector of the conductor,
• r^\hat{r} is the unit vector pointing from the current element to the point where the magnetic field is being calculated,
• rr is the distance from the current element to the point where the magnetic field is calculated.
Magnetic Field at the Center of a Current-Carrying Circular Loop
Now, let's derive an expression for the magnetic field at the center of a current-carrying circular loop using the Biot-Savart Law.
Step 1: Set up the problem
Consider a circular loop of radius RR carrying a steady current II. Let the center of the loop be point OO, where we want to calculate the magnetic field.
• The current flows along the loop, and we need to find the magnetic field at the center of the loop.
• We assume the loop lies in the xy-plane, and its center is at the origin (O).
Step 2: Apply the Biot-Savart Law
To find the total magnetic field at the center, we integrate the contributions from all differential current elements around the loop.
For a circular loop, the current element dld\mathbf{l} is tangential to the loop at every point. Let the position of the current element be described by the angle θ\theta around the loop.
The distance from any current element to the center is RR, and the angle between dld\mathbf{l} and the vector r^\hat{r} (which points radially from the element to the center) is always 90° because dld\mathbf{l} is tangential to the circle.
Thus, the cross-product dl×r^d\mathbf{l} \times \hat{r} will have a magnitude equal to dl×r^=dld\mathbf{l} \times \hat{r} = d\mathbf{l} (since sin⁡90∘=1\sin 90^\circ = 1).
Step 3: Calculate the magnetic field at the center
We now integrate over the entire loop. Each differential element contributes a small magnetic field at the center. Since the direction of the magnetic field due to each element is the same (perpendicular to the plane of the loop), the total magnetic field is the sum of all the individual contributions.
The Biot-Savart Law for the magnetic field at the center is:
dB=μ04πI dlR2dB = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l}}{R^2}
Since dl=R dθd\mathbf{l} = R \, d\theta, we can substitute this into the expression for dBdB:
dB=μ04πIR dθR2=μ04πI dθRdB = \frac{\mu_0}{4\pi} \frac{I R \, d\theta}{R^2} = \frac{\mu_0}{4\pi} \frac{I \, d\theta}{R}
To find the total magnetic field, integrate this expression around the full circle of the loop (i.e., θ\theta ranges from 0 to 2π2\pi):
B=∫02πμ04πI dθRB = \int_0^{2\pi} \frac{\mu_0}{4\pi} \frac{I \, d\theta}{R}
This simplifies to:
B=μ0I4πR∫02πdθB = \frac{\mu_0 I}{4\pi R} \int_0^{2\pi} d\theta
The integral of dθd\theta over the range 00 to 2π2\pi is just 2π2\pi, so:
B=μ0I4πR×2π=μ0I2RB = \frac{\mu_0 I}{4\pi R} \times 2\pi = \frac{\mu_0 I}{2R}
Final Expression:
Thus, the magnetic field at the center of a current-carrying circular loop is:
B=μ0I2RB = \frac{\mu_0 I}{2R}
Conclusion:
• The magnetic field at the center of a current-carrying circular loop of radius RR and current II is given by μ0I2R\frac{\mu_0 I}{2R}.
• This magnetic field is directed perpendicular to the plane of the loop (following the right-hand rule).