Magnetic field intensity (dH) produced at a point P by the differential current element I dl is given by:

(The angle between the element and the line joining P is α and the distance between P and the element is R) 

Explanation:

Magnetic Field Intensity Produced by a Differential Current Element

Definition: The magnetic field intensity (dH) at a point P due to a small current element Idl is a fundamental concept in electromagnetism. This is described by the Biot-Savart Law, which relates the magnetic field produced by a current-carrying conductor to the distance from the conductor and the angle between the current element and the position vector.

Biot-Savart Law: According to the Biot-Savart Law, the differential magnetic field intensity (dH) at a point P due to a differential current element Idl is given by:

\[\rm dH = \frac{I \, dl \, \sin \alpha}{4 \pi R^2}\]

Here,

• I is the current flowing through the differential element.
• dl is the length of the differential current element.
• R is the distance between the point P and the current element.
• α is the angle between the current element and the line joining the point P to the element.

Explanation of Correct Option:

The correct option is:

Option 2: \(\rm dH = \frac{Idl \sin \alpha}{4 \pi R^2}\)

This option correctly applies the Biot-Savart Law to describe the magnetic field intensity produced by the differential current element at point P. The law states that the magnetic field intensity is directly proportional to the current and the length of the current element, inversely proportional to the square of the distance, and depends on the sine of the angle between the element and the line joining the point P.

Derivation and Analysis:

To derive and understand why Option 2 is correct, let’s consider the Biot-Savart Law in more detail.

The Biot-Savart Law in vector form is given by:

\[\vec{dH} = \frac{I \, d\vec{l} \times \hat{r}}{4 \pi R^2}\]

Where:

• \(d\vec{l}\) is the differential length vector of the current element.
• \(\hat{r}\) is the unit vector in the direction from the current element to the point P.
• R is the distance between the point P and the current element.

The cross product \(d\vec{l} \times \hat{r}\) introduces the sine of the angle α between the current element and the line joining the point P to the element. Therefore, the magnitude of the magnetic field intensity can be written as:

\[\rm dH = \frac{I \, dl \, \sin \alpha}{4 \pi R^2}\]

This matches the expression given in Option 2, confirming its correctness.

Additional Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm dH = \frac{Idl \sin \alpha}{R^2}\)

This option does not include the factor \(4 \pi\) in the denominator, which is essential according to the Biot-Savart Law. The \(4 \pi\) factor arises from the integral form of the law and the symmetry of the magnetic field in three-dimensional space.

Option 3: \(\rm dH = \frac{Idl \sin \alpha}{2 \pi R^2}\)

This option includes a factor of \(2 \pi\) instead of \(4 \pi\). This discrepancy indicates an incorrect application of the Biot-Savart Law. The correct factor is \(4 \pi\), which accounts for the complete solid angle around the current element.

Option 4: \(\rm dH = \frac{Idl \sin \alpha}{4 \pi^2 R^2}\)

This option includes an additional \(\pi\) in the denominator, which is not justified by the Biot-Savart Law. The presence of \(4 \pi^2\) instead of \(4 \pi\) is incorrect and does not align with the standard formulation of the law.

Conclusion:

Understanding the Biot-Savart Law is crucial for accurately determining the magnetic field intensity produced by a current element. The correct expression for the magnetic field intensity at a point P due to a differential current element is:

\(\rm dH = \frac{Idl \sin \alpha}{4 \pi R^2}\)

This formulation highlights the dependence on the current, the length of the current element, the distance to the point, and the angle between the element and the line joining the point P. This fundamental law is a cornerstone in electromagnetism, providing insights into the behavior of magnetic fields around current-carrying conductors.