A small current element \( I\,d\vec{l} \) is placed at the origin, with \( d\vec{l} \) along the +x direction. At which of the following points is the magnetic field due to this element zero?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (2 m, 0, 0)
According to the Biot-Savart law, the field of a current element at a point is proportional to \( dl\sin\theta/r^{2} \). Here \( \theta \) is the angle between \( d\vec{l} \) and the displacement \( \vec{r} \) of the point. The field points along \( d\vec{l}\times\vec{r} \). The field can vanish only where \( \sin\theta=0 \), that is, at points lying on the line of the element itself.
Option (1) (0, 1 m, 0) — Incorrect. This point is on the y-axis, so \( \vec{r} \) is at right angles to \( d\vec{l} \) and \( \sin\theta=1 \). The field there is along \( \hat{\imath}\times\hat{\jmath}=\hat{k} \) and has its largest value for that distance.
Option (2) (0, 0, 1 m) — Incorrect. On the z-axis \( \theta \) is again \( 90^\circ \). The field is not zero and acts along \( \hat{\imath}\times\hat{k}=-\hat{\jmath} \).
Option (3) (1 m, 1 m, 0) — Incorrect. The line to this point makes \( 45^\circ \) with the x-axis, so \( \sin\theta=1/\sqrt{2} \). The field is smaller than on the y-axis but still not zero.
Option (4) (2 m, 0, 0) — Correct. The point lies on the x-axis, along \( d\vec{l} \). Here \( \theta=0 \), so \( \sin\theta=0 \) and \( d\vec{l}\times\vec{r}=0 \). So the element produces no field at this point.
NCERT (2026-27):
Magnetic Field Due to a Current Element, Biot-Savart Law – “(iv) There is an angle dependence in the Biot-Savart law which is not present in the electrostatic case. … the magnetic field at any point in the direction of dl … is zero. Along this line, θ = 0, sin θ = 0 and … |dB| = 0.”