Question

State Biot-Savart’s law. A current I flows in a conductor placed perpendicular to the plane of the paper. Indicate the direction of the magnetic field due to a small element dl at a point situated at a distance r from the element as shown in the figure.

Answer 1

Recall the Biot-Savart’s law and express it in scalar or vector form. To determine the direction of the magnetic field at point P, use Right-hand thumb rule. In right-hand thumb rule, curl fingers represent the direction of the magnetic field.

Complete answer:
We know that the current flowing through a differential length element dl of a conductor produces differential magnetic field at a distance r from it. The magnitude of this differential magnetic field is given by Biot-Savart’s law.
Biot-Savart’s law: According to Biot-Savart’s law the differential magnetic field dB produced by the current I flowing through a differential length element dl at point P is proportional to the current in the differential length element, length of element, sine of angle between unit vector of differential element and point P and it is inversely proportional to the square of distance between differential length element and point P.
Biot-Savart’s law can be expressed as follows,
$dB = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{I\,dl\sin \theta }}{{{r^2}}}$
Here, ${\mu _0}$ is the permeability of the free space.
The direction of magnetic field produced at point P due to current in the differential length element dl can be determined using Right-hand thumb rule.Right-hand thumb rule states that if we hold a conductor in our right hand such that the thumb represents the direction of current flow; then the curled fingers around the conductor represent the direction of the magnetic field at a point.

In the given figure, we have given that the current is flowing along the z-axis. Therefore, the Induced magnetic field at point P is towards the negative x-axis that is into the plane of paper.

Note: Using Biot-Savart’s law you can express the magnetic field at any point due to the current carrying conductor of any shape. It is not necessary that the conductor should be straight. Students should always use the right-hand thumb rule to determine the direction of magnetic current if only the direction of current is known. If direction of magnetic force is known then you can use Fleming’s left hand rule.