Ampere's Law and Its Applications

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 ?

An infinite wire placed along $z$-axis, has current $I_1$ in positive $z$-direction. A conducting rod placed in $xy$ plane parallel to y-axis has current $I_2$ in positive y-direction. The ends of the rod subtend angles of $+30°$ and $-60°$ at the origin with positive $x$-direction. The rod is at a distance $a$ from the origin. Find net force on the rod.

A $U$ shaped wire of mass $m$ and length $l$ is immersed with its two ends in mercury (see figure). The wire is in a homogeneous field of magnetic induction $B$. If a charge, that is, a current pulse $q=\int i\mathrm{d}t,$sent through the wire, the wire will jump up. Calculate, from the height $h$ that the wire reaches, the size of the charge or current pulse, assuming that the time of the current pulse is very small in comparison with the time of flight. Make use of the fact that impulse of force equals $\int F\mathrm{d}t,$ which equals $mv$. Evaluate $q$ for following details:
$B=0.1 \mathrm{Wb} {\mathrm{m}}^{-2}\mathrm{,} m=10 \mathrm{g}\mathrm{,}l=20 \mathrm{cm}$
$\text{\&} \mathit{h}=3 \mathrm{m}.\left[g=\mathrm{10} \mathrm{m} {\mathrm{s}}^{-2}\right]$

A rectangular loop of wire is oriented with the left corner at the origin, one edge along X-axis and the other edge along Y-axis as shown figure. A magnetic field is into the page and has a magnitude that is given by b=$\alpha \text{y}\text{where}\alpha \text{is }$constant. Find the total magnetic force on the loop if it carries current i.

Electric charge $q$ is uniformly distributed over a rod of length $l$. The rod is placed parallel to a long wire carrying a current $i$. The separation between the rod and the wire is $a$. Find the force needed to move the rod along its length with a uniform velocity $v$.

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 $\mathrm{C.}$

Figure shows a rectangular current-carrying loop placed $2 cm$ away from a long, straight, current-carrying conductor. What is the magnitude of the net force acting on the loop in ${10}^{-4}N$.

A positive charge is moving towards south in a space where magnetic field is pointing in the north direction. The moving charge will experience:

A wire carrying a current $I$ along the positive x-axis has length $L$. It is kept in a magnetic field $\overrightarrow{B}=\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-4\hat{\mathrm{k}} \right) \mathrm{T}$. The magnitude of the magnetic force acting on the wire is:

A straight wire AB of mass $40 \mathrm{g}$ and length $50 \mathrm{cm}$ is suspended by a pair of flexible leads in uniform magnetic field of magnitude $0.40 \mathrm{T}$ as shown in the figure. The magnitude of the current required in the wire to remove the tension in the supporting leads is _____ $\mathrm{A}$. (Take $\left.g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$.

A current carrying wire of length $l$ is first converted into a single loop ring and then into a $3$ loop coil. If magnetic field due to first ring is $B_1$, and due to second coil is $B_2$, at their respective centres, then the value of $\frac{B_1}{B_2}$ is

Two concentric circular coils with radii $1 \mathrm{cm}$ and $1000 \mathrm{cm}$ and number of turns $10$ and $200$ respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be _____ $\times {10}^{–8} \mathrm{H}$. (Take, ${\pi }^2=10$)

Two concentric and coplanar circular loops P and Q have their radii in the ratio $2:3$. Loop Q carries a current $9 \mathrm{A}$ in the anticlockwise direction. For the magnetic field to be zero at the common centre, loop P must carry

The net magnetic force on the loop $ABCD$ due the wire is

A closed curve encircles several conductors.The line integral $\int \overrightarrow{\mathrm{B}}.\overrightarrow{\mathrm{dl}}$ around this curve is $3.83\times {10}^{-7} \mathrm{T}-\mathrm{m}$. What is the net current in the conductors?

Only the current inside the amperian loop contributes in

The figure shows the cross-section of two long co-axial tubes carrying equal current in opposite direction at point 1 and 2, as shown in the figure then

A closed curve encircles several conductors. The line integral $\int \overrightarrow{B}.d\overrightarrow{l}$ around this curve is $3.83\times {10}^{-7} \mathrm{T}·\mathrm{m}$. What is the net current in the conductors?