An infinitely long wire,located on the z-axis | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The magnetic field at a distance $r$ from an infinitely long wire carrying current $I$ is given by $B = \frac{\mu_0 I}{2 \pi r}$.For a path elemen

Vedclass · Accepted Answer

$7 \mu_0 I / 24$

Vedclass · Answer

(A) The magnetic field at a distance $r$ from an infinitely long wire carrying current $I$ is given by $B = \frac{\mu_0 I}{2 \pi r}$.For a path element $d\vec{l}$ along a circular arc of radius $r$,$d\vec{l} = r d	heta$ and $\vec{B}$ is tangential to the path,so $\vec{B} \cdot d\vec{l} = B dl = \frac{\mu_0 I}{2 \pi r} (r d	heta) = \frac{\mu_0 I}{2 \pi} d	heta$.The line integral along the straight path from $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ corresponds to an angular sweep. The points lie on the line $y = a$. The distance from the $z$-axis to the line $y=a$ is $r = a / \cos	heta$.However,using the geometry of the path,the angle $	heta$ subtended at the origin changes from $	heta_1$ to $	heta_2$.For point $(-\sqrt{3} a, a, 0)$,$	an	heta_1 = \frac{|-\sqrt{3} a|}{a} = \sqrt{3} \implies 	heta_1 = \frac{\pi}{3}$.For point $(a, a, 0)$,$	an	heta_2 = \frac{a}{a} = 1 \implies 	heta_2 = \frac{\pi}{4}$.The integral is $\int \vec{B} \cdot d\vec{l} = \int_{-	heta_1}^{	heta_2} \frac{\mu_0 I}{2 \pi} d	heta = \frac{\mu_0 I}{2 \pi} (	heta_2 + 	heta_1) = \frac{\mu_0 I}{2 \pi} (\frac{\pi}{4} + \frac{\pi}{3}) = \frac{\mu_0 I}{2 \pi} (\frac{7\pi}{12}) = \frac{7 \mu_0 I}{24}$.

Similar Questions

Two concentric coils of $20$ turns each are placed in the same plane. Their radii are $30 \text{ cm}$ and $60 \text{ cm}$,and they carry $0.4 \text{ A}$ and $0.6 \text{ A}$ currents,respectively,in opposite directions. The magnetic induction at the centre in tesla is ....... .

The magnitude of the magnetic field at the centre of an equilateral triangular loop of side $1\,m$ which is carrying a current of $10\,A$ is:......$\mu T$ [Take $\mu _0 = 4\pi \times 10^{-7}\,NA^{-2}$]

$A$ circular coil of radius $R$ carries an electric current $I$. The magnetic field due to the coil at a point on the axis of the coil located at a distance $r$ from the centre of the coil,such that $r \gg R$,the magnetic field at that point is proportional to

Two infinitely long wires carrying currents of $8\,A$ and $6\,A$ are placed along the $X$-axis and $Y$-axis respectively. What is the magnetic field at point $P(0, 0, d)\,m$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module