Electromagnetic Induction · Physics · JEE Advanced

Consider a circuit consisting of a capacitor of capacitance C and a coil with N turns per unit length, cross sectional area S and length d, where $d^2 \gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is L. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

List-I contains four conducting loops lying in the $XY$ plane, as shown in the figures. The loops are rotating about $Z$ axis passing through the point $O$ with time period $T$ in clockwise direction.

The region $x>0$ contains a uniform magnetic field $B$ in the $+z$ direction. List-II contains the qualitative variation of the induced current $i(t)$ for each of these loops. Choose the option which describes the correct match between the entries in List-I to those in List-II.

List-I	List-II
(P)	(1)
(Q)	(2)
(R)	(3)
(S)	(4)
	(5)

A conducting square loop initially lies in the $X Z$ plane with its lower edge hinged along the $X$-axis. Only in the region $y \geq 0$, there is a time dependent magnetic field pointing along the $Z$-direction, $\vec{B}(t)=B_0(\cos \omega t) \hat{k}$, where $B_0$ is a constant. The magnetic field is zero everywhere else. At time $t=0$, the loop starts rotating with constant angular speed $\omega$ about the $X$ axis in the clockwise direction as viewed from the $+X$ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. $(V)$ in the loop as a function of time:

A region in the form of an equilateral triangle (in $x-y$ plane) of height $L$ has a uniform magnetic field $\vec{B}$ pointing in the $+z$-direction. A conducting loop $\mathrm{PQR}$, in the form of an equilateral triangle of the same height $L$, is placed in the $x-y$ plane with its vertex $\mathrm{P}$ at $x=0$ in the orientation shown in the figure. At $t=0$, the loop starts entering the region of the magnetic field with a uniform velocity $\vec{v}$ along the $+x$-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf $(E)$ in the loop as a function of the distance $(x)$ starting from $x=0$ ?

The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. $$I_1$$ and $$I_2$$ are the currents in the segments ab and cd. Then,

What is the advantage of this system?

What is the disadvantage of this system?

Which force causes the train to elevate upwards

A long solenoid of radius a and number of turns per unit length $$n$$ is enclosed by cylindrical shell of radius R, thickness $$d$$ $$(d < < R)$$ and length L. A variable current $$\mathrm{I}=\mathrm{I}_{0} \sin \omega t$$ flows through the coil. If the resistivity of the material of cylindrical shell is $$\mathrm{P}$$, find the induced current in the shell.

A conducting square loop of side $L$, mass $M$ and resistance $R$ is moving in the $X Y$ plane with its edges parallel to the $X$ and $Y$ axes. The region $y \geq 0$ has a uniform magnetic field, $\vec{B}=B_0 \widehat{k}$. The magnetic field is zero everywhere else. At time $t=0$, the loop starts to enter the magnetic field with an initial velocity $v_0 \hat{\jmath} \mathrm{~m} / \mathrm{s}$, as shown in the figure. Considering the quantity $K=\frac{B_0^2 L^2}{R M}$ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

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)

Two metallic rings A and B, identical in shape and size but having different resistivities $$\rho_A$$ and $$\rho_B$$, are kept on top of two identical solenoids as shown in the figure below. When current I is switched on in both the solenoids in identical manner, the rings A and B jump to heights $$h_A$$ and $$h_B$$, respectively, with $$h_A > h_B$$. The possible relation(s) between their resistivities and their masses $$m_A$$ and $$m_B$$ is (are)

A circular wire loop of radius R is placed in the xy plane centred at the origin O. A square loop of side a(a << R) having two turns is placed with its centre at z = $$\sqrt3$$R along the axis of the circular wire loop, as shown in the figure. The plane of the square loop makes an angle of 45$$^\circ$$ with respect to z-axis. If the mutual inductance between the loops is given by $${{{\mu _0}{a^2}} \over {{2^{p/2}}R}}$$, then the value of p is ___________.