The question asks about the dependence of the magnetic field's magnitude on the distance from the current element according to Biot-Savart law. Let's understand the Biot-Savart law and the correct option.

Biot-Savart law relates the magnetic field \(\vec{B}\) generated by a steady current to the current element and the distance from that element. The law is expressed for a small segment of current-carrying conductor as:

\(d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{r^3}\)

Where:

• \(\mu_0\) is the permeability of free space.
• \(I\) is the current flowing through the conductor.
• \(d\vec{l}\) is the differential length element of the conductor with direction along the current.
• \(\vec{r}\) is the position vector from the current element to the point of measurement.
• \(r\) is the magnitude of the position vector (distance from the current element to the point).

The cross-product \(d\vec{l} \times \vec{r}\) gives a direction perpendicular to both \(d\vec{l}\) and \(\vec{r}\), and its magnitude is maximum when \(d\vec{l}\) and \(\vec{r}\) are perpendicular.

Key observation: The magnetic field's magnitude is inversely proportional to the square of the distance from the current element, as evident from \(\frac{1}{r^2}\) in the formula since the cross product results in its magnitude being divided by \(r^3\) and further modified by \(r/r\) due to normalizing the directional aspect.

Analyzing the options:

• Directly proportional to the square of the distance: This is incorrect. The Biot-Savart law indicates an inverse relation.
• Directly proportional to the cube of the distance: This is also incorrect. The inverse square relation forms the basis of the law.
• Inversely proportional to the square of the distance: This is correct, as discussed above.
• Inversely proportional to the cube of the distance: Incorrect. The magnetic field decreases with distance, but only as an inverse square function.

Thus, according to the Biot-Savart law, the magnitude of the magnetic field due to a current element is inversely proportional to the square of the distance between the current element and the point where the field is calculated.