An element Δl = Δxi is placed at the origin and carries a large current I = 10A. The magnetic field on the y-axis

Question

An element \( \Delta l=\Delta x \hat{i}\) is placed at the origin and carries a large current \(\mathrm{I}=10 \mathrm{A}\). The magnetic field on the y-axis at a distance of \( 0.5 \mathrm{~m}\) from the elements \(\Delta \mathrm{x}\) of \( 1 \mathrm{~cm}\) length is :

(1) \( 4 \times 10^{-8} \mathrm{T}\)

(2) \(8 \times 10^{-8} \mathrm{T}\)

(3) \( 12 \times 10^{-8} \mathrm{T}\)

(4) \(10 \times 10^{-8} \mathrm{T}\)

Answer 1

Correct option is :  (1) \(4 \times 10^{-8} \mathrm{T}\) 

by biot-savart law

small magnetic field, \(\mathrm{dB}=\frac{\mu_0}{4 \pi} \cdot \frac{\mathrm{i}(\overrightarrow{\mathrm{dl}} \times \overrightarrow{\mathrm{r}})}{\mathrm{r}^3}=\frac{\mu_0}{4 \pi} \frac{\mathrm{i} \cdot \mathrm{dl} \sin \theta}{\mathrm{r}^2}\)

since element is very small, \(\mathrm{dl}=1 \mathrm{~cm}, \mathrm{r}=50 \mathrm{~cm}, \mathrm{i}=10 \mathrm{~A}, \sin\ \theta=1\)

\( \text { magnetic field }=\frac{\left(10^{-7}\right) \times(10)\left(10^{-2}\right)}{(0.5)^2}=4 \times 10^{-8} \ \mathrm{T} \)