Ampere's Circuital Law

Current I is following along the path ABCD, along the four edges of the cube (figure-a) creates a magnetic field in the centre of the cube of ${\text{B}}_0$. Find the magnetic field B created at the centre of the cube current I following along the path of the six edges ABCDHEA figure - b

Two long conductors are arranged as shown above to form overlapping cylinders, each of radius r, whose centres are separated by a distance d. Current of density J flows into the plane of the page along the shaded part of one conductor and an equal current flows out of the plane of the page along the shaded portion of the other, as shown. What are the magnitude and direction of the magnetic field at point A ?

A cylindrical conductor of radius $R$ carries a current along its length. The current density $J$, however, is not uniform over the cross-section of the conductor but is a function of the radius according to $J=br$, where $b$ is a constant. Then the expression for the magnetic field



$\left(B_1\right)$ at $r_1<R$ and
$\left(B_{\mathrm{2}}\right)$ at distance $r_2>R$, measured from the axis is:

Six wires of current ${\mathrm{I}}_1=\mathrm{1 }\mathrm{A},{\mathrm{I}}_2=\mathrm{2 }\mathrm{A},{\mathrm{I}}_3=\mathrm{3 }\mathrm{A},{\mathrm{I}}_4=\mathrm{1 }\mathrm{A},{\mathrm{I}}_5=\mathrm{5 }\mathrm{A}$ and ${\mathrm{I}}_6=\mathrm{4 }\mathrm{A}$ cut the page perpendicularly at the points $\mathrm{1,\; 2,\; 3,\; 4,\; 5}$ and $6$ respectively as shown in the figure. Find the value of integral $\oint \overrightarrow{B}.d\overrightarrow{l}$ around the circular path $\mathrm{C.}$

$PQRS$ is a square of side $l_0. \mathrm{A}, \mathrm{B}, \mathrm{C}$ and $D$ are four long current carrying wires kept perpendicular to the plane of paper as shown in the figure. $PA=QB=RC=SD=\frac{l_0}{\sqrt{5}}$. The magnitude of current in the wires $A, B, C$ and $D$ are $i_0, 2i_0, 8i_0$ and $4i_0$ respectively, the direction of currents is shown. It is given that ${\int }_P^Q{\overrightarrow{B}}_A·d\overrightarrow{l}=-9{\mu }_0$ $\text{Tesla meter}$ and ${\int }_R^{\mathrm{s}}{\overrightarrow{B}}_B·d\overrightarrow{l}=5{\mu }_0$ $\text{Tesla meter}$, where ${\overrightarrow{B}}_A$ and ${\overrightarrow{B}}_B$ are the magnetic fields due to the wires $A$ and $B$ respectively and paths considered for integral are shortest straight segments. The magnitude of $i_0$ is

Ampere's law is analogous to

A toroid of $n$ turns, mean radius $R$ and cross-sectional radius $a$ carries current $I.$ It is placed on a horizontal table taken as $xy$-plane. Its magnetic moment m _______

The figure shows a long coaxial cable in which a current $I$ flows down through the inner cylinder of radius $a$ and the same current flows back up through the outer cylinder of radius $2a$. The cylinders are insulated from each other and the current is uniformly distributed over the area of the cross-section in each cylinder. The strength of the magnetic field at a distance $\raisebox{1ex}{$3a$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ from the axis of the cable is

When the number of turns in a toroidal coil is doubled, then the value of magnetic flux density will become,

A homogeneous electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are pointing in the same direction. A proton is projected with its velocity parallel to $\overrightarrow{E}$. It will :

A long straight wire carrying a current $l_1$ is kept along the axis of a circular loop carrying a current $l_2.$ The long wire will

Properties of magnetic lines of force are:

Magnetic line of forces

Choose the incorrect option-

The two ends of $\mathrm{a}$ non-conducting spring of force constant $50{\mathrm{Nm}}^{-1}$ and unstretched length of $2 \mathrm{cm}$ are connected to the mid points of two straight parallel rods each of length $4 \mathrm{m}$. When $100 \mathrm{A}$ current is passed through each rod in the same direction, the work done on the spring in $\mathrm{mJ}$ is

STATEMENT-1: A uniformly distributed direct current flows through a solid long metallic cylinder along its length. The magnetic field produced by this current is only outside the cylinder.

STATEMENT- 2: A thin long cylindrical tube carrying uniformly distributed direct current along its length does not produce a magnetic field inside it. Moreover, a solid cylinder can be supposed to be made up of many thin cylindrical tubes.

Consider a toroid, having a circular cross-section of radius $\mathrm{b}$, major radius $\mathrm{R }\left(\mathrm{R}>>\mathrm{b}\right)$, having $\mathrm{N}$ turns and carrying current $\mathrm{I}$. Find the total energy stored in the toroid.

The average radius of a toroid made out of a nonmagnetic material is $0.1 \mathrm{m}$ and it has $500$ turns. If it carries $0.5 \mathrm{A}$ current, then the intensity of the magnetic field along its circular axis in $\text{tesla}$ will be

Ampere’s circuit law is given by

Ampere circuital equation for given problem is equal to