Biot-Savart law

Two concentric circular loops, one of radius $R$ and the other of radius $2R$, lie in the $xy$-plane with the origin as their common center, as shown in the figure. The smaller loop carries current $I_1$ in the anti-clockwise direction and the larger loop carries current $I_2$ in the clockwise direction, with $I_2 > 2I_1$. $\vec B(x, y)$ denotes the magnetic field at a point $(x, y)$ in the $xy$-plane. Which of the following statement(s) is(are) correct?

When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

Two infinitely long straight wires lie in the $xy$-plane along the lines $x = \pm R$. The wire located at $x = +R$ carries a constant current $I_1$ and the wire located at $x = -R$ carries a constant current $I_2$. A circular loop of radius $R$ is suspended with its centre at $(0, 0, \sqrt3R)$ and in a plane parallel to the $xy$-plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\hat j$ direction. Which of the following statements regarding the magnetic field $\vec B$ is (are) true?

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$. The magnitude of the magnetic field at the center of the loop is

A small circular loop of wire of radius a is located at the centre of a much larger circular wire loop of radius b. The two loops are in the same plane. The outer loop of radius b carries an alternating current $I=I_o\cos(\omega t)$. The emf induced in the smaller inner loop is nearly :

According to Bohr's theory, the time averaged magnetic field at the centre (i.e. nucleus) of a hydrogen atom due to the motion of electrons in the n$^{\text{th}}$ orbit is proportional to : (n = principal quantum number)

Two identical wires A and B, each of length '$l$', carry the same current I. Wire A is bent into a circle of radius R and wire B is bent to form a square of side '$a$'. If $B_A$ and $B_B$ are the values of magnetic field at the centres of the circle and square respectively, then the ratio $\dfrac{B_A}{B_B}$ is :

Two parallel wires in the plane of the paper are distance $X_0$ apart. A point charge is moving with speed $u$ between the wires in the same plane at a distance $X_1$ from one of the wires. When the wires carry current of magnitude $I$ in the same direction, the radius of curvature of the path of the point charge is $R_1$. In contrast, if the currents $I$ in the two wires have directions opposite to each other, the radius of curvature of the path is $R_2$. If $\dfrac{X_0}{X_1}=3$, the value of $\dfrac{R_1}{R_2}$ is

When $d\approx a$ but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height $h$ above the loop. In that case

Consider $d\gg a$, and the loop is rotated about its diameter parallel to the wires by 30° from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

Three straight parallel current carrying conductors are shown in the figure. The force experienced by the middle conductor of length 25 cm is :

Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field $B_1$, at its centre when a current I passes through it. The second wire is bent into a coil with three identical loops adjacent to each other and produces magnetic field $B_2$ at the centre of the loops when current I/3 passes through it. The ratio $B_1:B_2$ is :

A square frame of side 10 cm and a long straight wire carrying current 1 A are in the plane of the paper. Starting from close to the wire, the frame moves towards the right with a constant speed of 10 ms$^{-1}$ (see figure). The e.m.f induced at the time the left arm of the frame is at $x=10$ cm from the wire is :

A circular wire loop of radius $R$ is placed in the $x$-$y$ plane centered at the origin $O$. A square loop of side $a$ $(a\ll R)$ having two turns is placed with its center at $z=\sqrt3R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45° with respect to the $z$-axis. If the mutual inductance between the loops is given by $\dfrac{\mu_0a^2}{2^{p/2}R}$, then the value of $p$ is

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

A long insulated copper wire is closely wound as a spiral of '$N$' turns. The spiral has inner radius '$a$' and outer radius '$b$'. The spiral lies in the X-Y plane and a steady current '$I$' flows through the wire. The Z-component of the magnetic field at the center of the spiral is

Six point charges, each of the same magnitude $q$, are arranged in different manners as shown in Column II. In each case, a point M and a line PQ passing through M are shown. Let $E$ be the electric field and $V$ be the electric potential at M (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line PQ. Let $B$ be the magnetic field at M and $\mu$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.

A steady current $I$ goes through a wire loop PQR having shape of a right angle triangle with PQ $=3x$, PR $=4x$ and QR $=5x$. If the magnitude of the magnetic field at P due to this loop is $k\left(\dfrac{\mu_0I}{48\pi x}\right)$, find the value of $k$.

Two wires each carrying a steady current $I$ are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.