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  • A circular loop of radius 'R' is placed in the xy-plane with its centre at the origin. It carries a current I in the clockwise direction. Find the magnetic field at a point (0,0,h) along th

A circular loop of radius 'R' is placed in the xy-plane with its centre at the origin. It carries a current I in the clockwise direction. Find the magnetic field at a point (0,0,h) along the z-axis.

Option: 1

\frac{\mu_0 I^2 R^2 h}{2\left(R^2+h^2\right)^{3 / 2}}


Option: 2

\frac{\mu_0 I R^2}{2\left(R^2-h^2\right)^{3 / 2}}


Option: 3

\frac{\mu_0 I R^2}{3\left(R^2+h^2\right)^{3 / 2}}


Option: 4

\frac{\mu_0 I R^2}{2\left(R^2-h^2\right)^{1 / 2}}


Answers (1)

best_answer

According to Biot-Savart law

B=\frac{\mu_0 I}{4 \pi} \cdot \int \frac{d \vec{I} \times \vec{r}}{r^3}

Since d \vec{I}and \vec{r} are perpendicular to the z-axis.

d \vec{I} \times \vec{r}=d I \cdot r \cdot \sin \phi \cdot \hat{\phi} .

\text { So } B=\frac{\mu_0 I}{4 \pi} \cdot \int_0^{2 \pi} \frac{J R \cdot \sqrt{R^2+h^2} \cdot h / \sqrt{R^2+h^2}}{\left(R^2+h^2\right)^{3 / 2}} \cdot R \phi \phi \hat{\phi}

\begin{aligned} & =\frac{\mu_0 I^2 R^2 h}{4 \pi\left(R^2+h^2\right)^{3 / 2}} \cdot \int_0^{2 \pi} d \phi \cdot \hat{\phi} \\ & =\frac{\mu_0 I^2 R^2 h}{4 \pi\left(R^2+h^2\right)^{3 / 2}} \cdot 2 \pi \hat{\phi} \\ & =\frac{\mu_0 I^2 R^2 h}{2\left(R^2+h^2\right)^{3 / 2}} \hat{\phi} \end{aligned}

Posted by

Gautam harsolia

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