Biot-Savart law

A hollow, right circular cone of base radius $R$ and height $h$, with its tip at the origin is rotating about the $Z$-axis with an angular velocity $\omega$, as shown in the figure. The cone carries a total charge $Q$ uniformly distributed on its curved surface. The magnitude of magnetic field at a point $(0,0,z)$, where $z\gg R$ and $z\gg h$, is $\dfrac{n\mu_0}{4\pi}\dfrac{QR^2\omega}{z^3}$. The value of $n$ is:

Two identical long current carrying wires are bent into the shapes shown in the following figures. If the magnitude of magnetic fields at the centres P and Q of a semicircular arc are $B_1$ and $B_2$ respectively, then the ratio $\dfrac{B_1}{B_2}$ is ________.

A circular current loop of radius $R$ is placed inside square loop of side length $L$ ($L\gg R$) such that they are co-planar and their centers coincide. The permeability of free space is $\mu_0$. The mutual inductance between circular loop and square loop is ________.

Using Bohr's model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the 2nd and 4th orbits of hydrogen atom ________.

A small cube of side 1 mm is placed at the centre of a circular loop of radius 10 cm carrying a current of 2 A. The magnetic energy stored inside the cube is $\alpha\times10^{-14}$ J. The value of $\alpha$ is ________.

($\mu_o=4\pi\times10^{-7}$ Tm/A, $\pi=3.14$)

A current of 30 A each flows in opposite directions in two conducting wires, placed parallel to each other at a distance of 8 cm. The magnetic field at the mid point between the two wires is ________ μT.

$\left(\dfrac{\mu_o}{4\pi}=10^{-7}\text{ N/A}^2\right)$

An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B}\cdot\vec{dl}$ along a straight line from the point $(-\sqrt{3}a, a, 0)$ to $(a, a, 0)$ is given by

[$\mu_0$ is the magnetic permeability of free space.]

A rectangular conducting loop of length 4 cm and width 2 cm is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\dfrac{\sqrt3}{2}\hat x + \dfrac12\hat y$ with a constant speed v. The wire is carrying a steady current $I = 10$ A in the positive $x$-direction. A current of $10\ \mu$A flows through the loop when it is at a distance $d = 4$ cm from the wire. If the resistance of the loop is $0.1\ \Omega$, then the value of v is _____ $\mathrm{m\,s^{-1}}$.

[Given: The permeability of free space $\mu_0 = 4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$]

Which one of the following options represents the magnetic field $\vec B$ at O due to the current flowing in the given wire segments lying on the $xy$ plane?

The magnetic field at the centre of a circular coil of radius $r$, due to current $I$ flowing through it, is $B$. The magnetic field at a point along the axis at a distance $\dfrac r2$ from the centre is :

Two long parallel conductors $S_1$ and $S_2$ are separated by a distance 10 cm and carrying currents of 4A and 2A respectively. The conductors are placed along $x$-axis in X-Y plane. There is a point P located between the conductors (as shown in figure).

A charge particle of $3\pi$ coulomb is passing through the point P with velocity $\vec v=\left(2\hat i+3\hat j\right)$ m/s; where $\hat i$ & $\hat j$ represents unit vector along $x$ & $y$ axis respectively.

The force acting on the charge particle is $4\pi\times10^{-5}\left(-x\hat i+2\hat j\right)$ N. The value of $x$ is :

Two long current carrying conductors are placed parallel to each other at a distance of 8 cm between them. The magnitude of magnetic field produced at mid-point between the two conductors due to current flowing in them is 300 $\mu$T. The equal current flowing in the two conductors is :

A coil of n number of turns wound tightly in the form of a spiral with inner and outer radii $r_1$ and $r_2$ respectively. When a current of strength I is passed through the coil, the magnetic field at its centre will be :

A small square loop of wire of side $l$ is placed inside a large square loop of wire L ($L\gg l$). Both loops are coplanar and their centres coincide at point O as shown in figure. The mutual inductance of the system is :

The electric current in a circular coil of 2 turns produces a magnetic induction $B_1$ at its centre. The coil is unwound and is rewound into a circular coil of 5 turns and the same current produces a magnetic induction $B_2$ at its centre. The ratio of $\dfrac{B_2}{B_1}$ is

$B_X$ and $B_Y$ are the magnetic fields at the centre of two coils X and Y respectively each carrying equal current. If coil X has 200 turns and 20 cm radius and coil Y has 400 turns and 20 cm radius, the ratio of $B_X$ and $B_Y$ is :

The magnetic field at the center of current carrying circular loop is $B_1$. The magnetic field at a distance of $\sqrt3$ times radius of the given circular loop from the center on its axis is $B_2$. The value of $B_1/B_2$ will be

A closely wounded circular coil of radius 5 cm produces a magnetic field of $37.68\times10^{-4}$ T at its center. The current through the coil is _______ A.

[Given, number of turns in the coil is 100 and π = 3.14]

In the following loop, magnitude of the magnetic field produced at the centre (O) of the loop is : (Assume AB and CD are infinitely extended):

A long straight wire carries a current, $I = 2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time $t = 0$, the rod starts moving on the rails with a speed $v = 3.0$ m/s (see the figure).

A resistor $R = 1.4\ \Omega$ and a capacitor $C_o = 5.0\ \mu$F are connected in series between the rails. At time $t = 0$, $C_o$ is uncharged. Which of the following statement(s) is(are) correct? [$\mu_0 = 4\pi \times 10^{-7}$ SI units. Take $\ln 2 = 0.7$]