JEE Main 2026 — Electromagnetism

A circular coil of radius $2$ cm and $125$ turns carries a current of $1$ A. The coil is placed in a uniform magnetic field of magnitude $0.4$ T. The axis of the coil makes an angle of $30°$ with the direction of the magnetic field. The torque acting on the coil is $\alpha \times 10^{-4}$ N.m. The value of $\alpha$ is ______. ($\pi=3.14$)

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 $\alpha \times 10^{-14}$ J. The value of $\alpha$ is _______. ($\mu_o = 4\pi \times 10^{-7}$ Tm/A, $\pi = 3.14$)

$1\,\mu$C charge moving with velocity $\vec{v} = \left(\hat{i} - 2\hat{j} + 3\hat{k}\right)$ m/s in the region of magnetic field $\vec{B} = \left(2\hat{i} + 3\hat{j} - 5\hat{k}\right)$ T. The magnitude of force acting on it is $\sqrt{\alpha} \times 10^{-6}$ N. The value of $\alpha$ is _______.

A short bar magnet placed with its axis at $30^{\circ}$ with an external field of 800 Gauss, experiences a torque of $0.016 \mathrm{~N}. \mathrm{m}$. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \mathrm{~J}$. The value of $\alpha$ is $\_\_\_\_$.

A series LCR circuit with $R = 20\ \Omega$, $L = 1.6\text{ H}$ and $C = 40\ \mu\text{F}$ is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is _______ $\Omega$.

A charged particle moves in a uniform magnetic field. If the velocity is perpendicular to the field, the path of the particle is:

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^{\circ}$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is $\_\_\_\_$ mV. (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$)

When a coil is placed in a time dependent magnetic field the power dissipated in it is $P$. The number of turns, area of the coil and radius of the coil wire are $N$, $A$ and $r$ respectively. For a second coils number of turns, area of the coil and radius of the coil wire are $2N$, $2A$ and $3r$ respectively. When the first coil is replaced with second coil the power dissipated in it is $\sqrt{2}\,\alpha P$. The value of $\alpha$ is _______.

A 20 m long uniform copper wire held horizontally is allowed to fall under the gravity ($g=10 \mathrm{~m} / \mathrm{s}^{2}$) through a uniform horizontal magnetic field of 0.5 Gauss perpendicular to the length of the wire. The induced EMF across the wire when it travells a vertical distance of 200 m is $\_\_\_\_$ mV.

Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current $I=10 \sin (\omega \mathrm{t}) \mathrm{A}$, where $\omega=1000 \mathrm{rad}. / \mathrm{s}$. A circular conducting loop $(B)$ of radius 1 cm coaxially slided through the solenoid at a speed $v=1 \mathrm{~cm} / \mathrm{s}$. The r.m.s. current through the loop when the coil $B$ is inserted 10 cm inside the solenoid is $\alpha / \sqrt{2} \mu \mathrm{~A}$. The value of $\alpha$ is $\_\_\_\_$. [Resistance of the loop $=10 \Omega$ ]

Three identical coils $C_{1}, C_{2}$ and $C_{3}$ are closely placed such that they share a common axis. $C_{2}$ is exactly midway. $C_{1}$ carries current $I$ in anti-clockwise direction while $C_{3}$ carries current $I$ in clockwise direction. An induced current flows through $C_{2}$ will be in clockwise direction when

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. If the resistance of the total circuit is $2 \Omega$ then the force needed to move the rod towards right with constant speed (v) of $1.5 \mathrm{~m} / \mathrm{s}$ is $\_\_\_\_$ N. 

An inductor of $10$ mH, capacitor of $0.1\ \mu$F and a resistor of $100\ \Omega$ are connected in series across an $a.c$ power supply $220$ V, $70$ Hz. The power factor of the given circuit is $0.5$. The difference in the inductive reactance and capacitance reactance is $\sqrt{3}\alpha\ \Omega$. The value of $\alpha$ is _____.

Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V=5 \sin (100 t)$ The values of $L$ and $R$ are shown in the figure. The capacitance of the capacitor (C) used is $\_\_\_\_$ $\mu \mathrm{F}$. 

Inductance of a coil with $10^{4}$ turns is 10 mH and it is connected to a dc source of 10 V with internal resistance of $10 \Omega$. The energy density in the inductor when the current reaches $\left(\frac{1}{e}\right)$ of its maximum value is $\alpha \pi \times \frac{1}{e^{2}} \mathrm{~J} / \mathrm{m}^{3}$. The value of $\alpha$ is $\_\_\_\_$. ($\mu_{0}=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}$).

The current passing through a conducting loop in the form of equilateral triangle of side $4 \sqrt{3} \mathrm{~cm}$ is 2 A. The magnetic field at its centroid is $\alpha \times 10^{-5} \mathrm{~T}$. The value of $\alpha$ is $\_\_\_\_$. (Given : $\mu_{\mathrm{o}}=4 \pi \times 10^{-7}$ SI units)

For an electromagnetic wave propagating through vacuum, $\vec{k}$, $\vec{E}$ and $\omega$ represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by :

A monochromatic source of light operating at $15$ kW emits $2.5\times 10^{22}$ photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ________. (Take $h=6.6\times 10^{-34}$ J·s and $c=3\times 10^8$ m/s).

A magnetic field vector in an electromagnetic wave is represented by $\vec{B}=B_0\sin\left(2\pi vt-\dfrac{2\pi x}{\lambda}\right)\hat{j}$. Its associated electric field vector is ______.

A plane electromagnetic wave is moving in free space with velocity $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ and its electric field is given as $\vec{E}=54 \sin (k z-\omega t) \hat{j} \mathrm{~V} / \mathrm{m}$, where $\hat{j}$ is the unit vector along $y$-axis. The magnetic field vector $\vec{B}$ of the wave is:

Match List - I with List - II. $\begin{array}{l} \text{List - I} & \text{List - II} \\ \text{Relation} & \text{Law} \\ \text{A. } \oint \vec{E} \cdot \overrightarrow{d l}=-\frac{d}{d t} \oint \vec{B} \cdot \overrightarrow{d a} & \text{I. Ampere's circuital law} \\ \text{B. } \oint \vec{B} \cdot \overrightarrow{d l}=\mu_{0}\left(I+\epsilon_{0} \frac{d \phi_{E}}{d t}\right) & \text{II. Faraday's laws of} \\ & \text{electromagnetic induction} \\ \text{C. } \oint \vec{E} \cdot \overrightarrow{d a}=\frac{1}{\epsilon_{0}} \int_{\mathrm{V}} \rho \mathrm{dv} & \text{III. Ampere - Maxwell law} \\ \text{D. } \oint \vec{B} \cdot \overrightarrow{d l}=\mu_{0} I & \text{IV. Gauss's law of electrostatics} \end{array}$ Choose the correct answer from the options given below :

The equation of the electric field of an electromagnetic wave propagating through free space is given by : $E=\sqrt{377} \sin \left(6.27 \times 10^{3} t-2.09 \times 10^{-5} x\right) \mathrm{N} / \mathrm{C}$ The average power of the electromagnetic wave is $\left(\frac{1}{\alpha}\right) \mathrm{W} / \mathrm{m}^{2}$. The value of $\alpha$ is $\_\_\_\_$ (Take $\sqrt{\frac{\mu_{0}}{\varepsilon_{o}}}=377$ in SI units)

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, $\sigma=10 \mathrm{mho} / \mathrm{m}$. The ratio of maximum conduction current density to maximum displacement current density is $\_\_\_\_$. $\left[\text{Take }\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right]$

A point charge $q=1 \mu \mathrm{C}$ is located at a distance 2 cm from one end of a thin insulating wire of length 10 cm having a charge $Q=24 \mu \mathrm{C}$, distributed uniformly along its length, as shown in figure. Force between $q$ and wire is $\_\_\_\_$ N. (Use : $\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^{9} \mathrm{~N}. \mathrm{m}^{2} / \mathrm{C}^{2}$) 

Two shorts dipoles $(A, B), A$ having charges $\pm 2 \mu \mathrm{C}$ and length 1 cm and $B$ having charges $\pm 4 \mu \mathrm{C}$ and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point $P$, equi-distant from the centres of both dipoles is $\_\_\_\_$ N/C. 

An electromagnetic wave travels in free space along the $x$-direction. At a particular point in space and time, $\vec{B} = 2 \times 10^{-7} \hat{j}$ T is associated with this wave. The value of corresponding electric field $\vec{E}$ at this point is _______ V/m.

A metal rod of length $L$ rotates about one end at origin with a uniform angular velocity $\omega$. The magnetic field radially falls off as $B(r) = B_0 e^{-\lambda r}$; $\lambda$ being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is :

A three coulomb charge moves from the point $(0, -2, -5)$ to the point $(5, 1, 2)$ in an electric field expressed as $\vec{E} = 2x\hat{i} + 3y^2\hat{j} + 4\hat{k}$ N/C. The work done in moving the charge is _______ J.

A conducting circular loop of area $1.0 \mathrm{~m}^{2}$ is placed perpendicular to a magnetic field which varies as $B=\sin (100 t)$ Tesla. If the resistance of the loop is $100 \Omega$, then the average thermal energy dissipated in the loop in one period is $\_\_\_\_$ J.

An inductor stores 16 J of magnetic field energy and dissipates 32 W of thermal energy due to its resistance when an a.c. current of 2 A (rms) and frequency 50 Hz flows through it. The ratio of inductive reactance to its resistance is $\_\_\_\_$. $(\pi=3.14)$

The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \mu \mathrm{~T}$. The magnetic field at a distance $x=\sqrt{3} R$ on its axis from the centre is $\_\_\_\_$ $\mu \mathrm{T}$.

A conducting circular loop is rotated about its diameter at a constant angular speed of $100 \mathrm{rad} / \mathrm{s}$ in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^{\circ}$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is $\_\_\_\_$ mm. (Take $\pi=\frac{22}{7}$)

The electric current in the circuit is given as $i=i_{\mathrm{o}}(t / T)$. The r.m.s current for the period $t=0$ to $t=T$ is $\_\_\_\_$.

A displacement current of $4.0$ A can be set up in the space between two parallel plates of $6$ $\mu$F capacitor. The rate of change of potential difference across the plates of the capacitor is nearly $\alpha \times 10^6$ V/s. The value of $\alpha$ is __________.

Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by 15 cm length of wire $Q$ is $\_\_\_\_$.  ($\mu_{\mathrm{o}}=4 \pi \times 10^{-7} \mathrm{~T}. \mathrm{m} / \mathrm{A}$)

Three charges $+2 q,+3 q$ and $-4 q$ are situated at $(0,-3 a),(2 a, 0)$ and $(-2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is $\_\_\_\_$.

Two metal plates $(A, B)$ are kept horizontally with separation of $\left(\dfrac{12}{\pi}\right)$ cm, with plate A on the top. An atomizer jet sprays oil (density $1.5$ g/cm$^3$) droplets of radius $1$ mm horizontally. All oil droplets carry a charge $5$ nC. The potentials $V_A$ and $V_B$ are required on plates A and B respectively in order to ensure the droplets do not descend. The values of $V_A$ and $V_B$ are _______. (Neglect the air resistance to the droplets and take $g = 10$ m/s$^2$)

A LCR series circuit driven with $E_{rms} = 90$ V at frequency $f_d = 30$ Hz has resistance $R = 80\,\Omega$, an inductance with inductive reactance $X_L = 20.0\,\Omega$ and capacitance with capacitive reactance $X_C = 80.0\,\Omega$. The power factor of the circuit is _______.

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant $k=3$ and permeability of $\mu=2 \mu_{0}$, is ($\mu_{0}=$ permeability of vacuum)

A circular current loop of radius $R$ is placed inside square loop of side length $L$ $(L >> R)$ 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 _______.

$X P Q Y$ is a vertical smooth long loop having a total resistance $R$ where $P X$ is parallel to $Q Y$ and separation between them is $l$. A constant magnetic field $B$ perpendicular to the plane of the loop exists in the entire space. A rod $C D$ of length $L(L>l)$ and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}.(\mathrm{g}=$ acceleration due to gravity) 

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: In electrostatics, a conductor does not store any net charge inside. Reason R: Inside the capacitor (with no dielectric medium), the free charge carriers, if placed between the plates of capacitor, experience force and drift. Choose the correct answer from the options given below

A $30$ cm long solenoid has $10$ turns per cm and area of $5$ cm$^2$. The current through the solenoid coil varies from $2$ A to $4$ A in $3.14$ s. The e.m.f. induced in the coil is $\alpha\times 10^{-5}$ V. The value $\alpha$ is ________.

A laser beam has intensity of $4.0 \times 10^{14} \mathrm{~W} / \mathrm{m}^{2}$. The amplitude of magnetic field associated with beam is $\_\_\_\_$ T. (Take $\epsilon_{\mathrm{o}}=8.85 \times 10^{-12} \mathrm{C}^{2} / \mathrm{Nm}^{2}$ and $\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$)

Two charges $7 \mu \mathrm{C}$ and $-2 \mu \mathrm{C}$ are placed at $(-9,0,0) \mathrm{cm}$ and $(9,0,0) \mathrm{cm}$ respectively in an external field $E=\frac{\mathrm{A}}{r^{2}} \hat{r}$, where $A=9 \times 10^{5} \mathrm{~N} / \mathrm{C}. \mathrm{m}^{2}$. Considering the potential at infinity is 0, the electrostatic energy of the configuration is $\_\_\_\_$ J.

A thin half ring of radius $35$ cm is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is $100$ V/m, then the value of $Q$ is _______ nC. ($\epsilon_o = 8.85 \times 10^{-12}$ C$^2$/Nm$^2$ and $\pi = 3.14$)

Two resistors $2 \Omega$ and $3 \Omega$ are connected in the gaps of bridge as shown in figure. The null point is obtained with the contact of jockey at some point on wire $X Y$. When an unknown resistor is connected in parallel with $3 \Omega$ resistor, the null point is shifted by 22.5 cm toward $Y$. The resistance of unknown resistor is $\_\_\_\_$ $\Omega$. 

A parallel plate air capacitor has a capacitance $C$. When it is half filled as show in figure with a dielectric constant $K = 5$, the percentage increase in the capacitance is _______. 

A long straight wire carries a current of 10 A. The magnetic field at a distance of 0.1 m from the wire is:

Five positive charges each having charge $q$ are placed at the vertices of a pentagon as shown in the figure. The electric potential $(V)$ and the electric field $(\vec{E})$ at the center $O$ of the pentagon due to these five positive charges are: 

When an external resistance of $5\ \Omega$ is connected across terminals of a cell, a current of $0.25\text{ A}$ flows through it. When the $5\ \Omega$ resistor is replaced by a $2\ \Omega$ resistor, a current of $0.5\text{ A}$ flows through it. The internal resistance of the cell is _______ $\Omega$.

A square loop of side $2$ cm is placed in a time varying magnetic field with magnitude as $B = 0.4\sin(300t)$ Tesla. The normal to the plane of loop makes an angle of $60°$ with the field. The maximum induced emf produced in the loop is __________ mV.

For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6 \Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is $\_\_\_\_$ $\Omega$.

A regular hexagon is formed by six wires each of resistance $r \Omega$ and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

A point light source emits E.M. waves in free space. A detector, placed at a distance of $L$ m, measures the intensity as $I_o$. The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as $45°$. The measured intensity at new location will be _______.

The space between the plates of a parallel plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants $K_{1}=2, K_{2}=3$ and $K_{3}=5$ (as shown in figure). If new capacitance is $\frac{n}{3} C$ then the value of $n$ is $\_\_\_\_$. 

A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is : A. maximum at either ends of the conductor and minimum at the midpoint B. maximum at the axis of the conductor C. minimum at the surface of the conductor D. minimum at the axis of the conductor E. same at all points in the cross-section of the conductor Choose the correct answer from the options given below :

A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws $25 \%$ more charge from the battery. The dielectric constant of mica is $\_\_\_\_$.

The electric field in a plane electromagnetic wave is given by : $E_{y}=69 \sin \left[0.6 \times 10^{3} x-1.8 \times 10^{11} t\right] \mathrm{V} / \mathrm{m}$. The expression for magnetic field associated with this electromagnetic wave is $\_\_\_\_$ T.

The charged particle moving in a uniform magnetic field of $(3\hat{i} + 2\hat{j})$ T has an acceleration $\left(4\hat{i} - \dfrac{x}{2}\hat{j}\right)$ m/s$^2$. The value of $x$ is __________.

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 __________ $\mu$T. $\left(\dfrac{\mu_0}{4\pi} = 10^{-7}\text{ N/A}^2\right)$

An a.c. source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current is observed as $I$. Now the frequency of the source is changed to $\omega/4$, (keeping the voltage unchanged) the current is found to be $I/3$. The ratio of resistance to reactance at frequency $\omega$ is

The electric potential as a function of $x, y$ is given by $V = 5(x^2 - y^2)$ V. The electric field at a point $(2, 3)$ m is __________ V/m.

A capacitor $C$ is first charged fully with potential difference of $V_{0}$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance $L$. In $t \mathrm{~s} \ 25 \%$ of the initial energy in the capacitor is transferred to the inductor. The value of $t$ is $\_\_\_\_$ s.

For the series $L C R$ circuit connected with $220 \mathrm{~V}, 50 \mathrm{~Hz}$ a.c source as shown in the figure, the power factor is $\frac{\alpha}{10}$. The value of $\alpha$ is $\_\_\_\_$. 

A sphere of capacitance $100$ pF is charged to a potential of $100$ V. Another identical uncharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is $\alpha \times 10^{-7}$ J. The value of $\alpha$ is __________. (combined capacitance of spheres is $200$ pF)

Two charged conducting spheres $S_1$ and $S_2$ of radii $8$ cm and $18$ cm are connected to each other by a wire. After equilibrium is established, the ratio of electric fields on $S_1$ and $S_2$ spheres are $E_{S_1}$ and $E_{S_2}$ respectively. The value of $\dfrac{E_{S_1}}{E_{S_2}}$ is _______.

The dimensional formula of $\dfrac{1}{2}\epsilon_0 E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^a L^b T^c$. The value of $2a - b + c = $ _______.

An inductor of inductance $10$ mH having resistance of $100\,\Omega$ is connected to battery of E.M.F. $1.0$ V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is $2$ mA and $4$ mA is _______. 

Figure shows the circuit that contains three resistances ($9 \Omega$ each) and two inductors (4 mH each). The reading of ammeter at the moment switch $K$ is turned ON, is $\_\_\_\_$ A. 

There are three co-centric conducting spherical shells $A, B$ and $C$ of radii $a, b$ and $c$ respectively $(c>b>a)$ and they are charged with charge $q_{1}, q_{2}$ and $q_{3}$ respectively. The potentials of the spheres $A, B$ and $C$ respectively, are :

The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x, t)=25 \sin \left(2.0 \times 10^{15} t-10^{7} x\right) \hat{n}$ then the refractive index of the medium is $\_\_\_\_$. (All given measurement are in SI units)

The electric field of a plane electromagnetic wave, travelling in an unknown nonmagnetic medium is given by, $E_{\mathrm{y}}=20 \sin \left(3 \times 10^{6} x-4.5 \times 10^{14} \mathrm{t}\right) \mathrm{V} / \mathrm{m}$ (where $x, \mathrm{t}$ and other values have S.I. units). The dielectric constant of the medium is $\_\_\_\_$ (speed of light in free space is $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$)

An electromagnetic wave travelling in $x$-direction is described by field equation $E_y = 300\sin\omega\left(t - \dfrac{x}{c}\right)$. If the electron is restricted to move in $y$-direction only with speed of $1.5 \times 10^6$ m/s then ratio of maximum electric and magnetic forces acting on the electron is _______.

Match the LIST-I with LIST-II \(\begin{array}{|l|l|l|l|} \hline & \text{List-I} & & \text{List-II} \\ \hline \text{A.} & \text{Radio-wave} & \text{I.} & \text{is produced by Magnetron valve} \\ \hline \text{B.} & \text{Micro-wave} & \text{II.} & \text{due to change in the vibrational modes of atoms} \\ \hline \text{C.} & \text{Infrared-wave} & \text{III.} & \text{due to inner shell electrons moving from higher to lower energy level} \\ \hline \text{D.} & \text{X-ray} & \text{IV.} & \text{due to rapid acceleration of electrons} \\ \hline \end{array}\) Choose the correct answer from the options given below:

A point charge of $10^{-8} \mathrm{C}$ is placed at origin. The work done in moving a point charge $2 \mu \mathrm{C}$ from point $A(4,4,2) \mathrm{m}$ to $B(2,2,1) \mathrm{m}$ is $\_\_\_\_$ J. $\left(\frac{1}{4 \pi \epsilon_{o}}=9 \times 10^{9}\right.$ in SI units $)$

A wire of uniform resistance $\lambda \Omega / \mathrm{m}$ is bent into a circle of radius $r$ and another piece of wire with length $2 r$ is connected between points $A$ and $B(A O B)$ as shown in figure. The equivalent resistance between points $A$ and $B$ is $\_\_\_\_$ $\Omega$. 

The total length of potentiometer wire AB is 50 cm in the arrangement as shown in figure. If $P$ is the point where the galvanometer shows zero reading then the length $A P$ is $\_\_\_\_$ cm. 

A moving coil of galvanometer when shunted with $2\,\Omega$ resistance gives a full scale deflection for a current of $500$ mA. When a resistance of $470\,\Omega$ is connected in series it gives a full scale deflection for $10$ V potential applied on it. The value of resistance of galvanometer coil is __________ $\Omega$.

Refer to the figure given below. The values of $I_1$, $I_2$ and $I_3$ are __________. 

The charge stored by the capacitor $C$ in the given circuit in the steady state is $\_\_\_\_$ $\mu \mathrm{C}$. 

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec{E}_1=E_0\hat{x}$. If another electric field $\vec{E}_2=2E_0(\hat{y}+\hat{z})$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?

A current carrying solenoid is placed vertically and a particle of mass $m$ with charge $Q$ is released from rest. The particle moves along the axis of solenoid. If $g$ is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

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 _______. 

The equivalent resistance between the points $A$ and $B$ in the following circuit is $\frac{x}{5} \Omega$. The value of $x$ is $\_\_\_\_$. 

A particle having charge $10^{-9}$ C moving in $x$-$y$ plane in fields of $0.4\hat{j}$ N/C and $4 \times 10^{-3} \hat{k}$ T experiences a force of $(4\hat{i} + 2\hat{j}) \times 10^{-10}$ N. The velocity of the particle at that instant is _______ m/s.

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Consider a ferromagnetic material : Assertion (A) : The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains. Reason (R): At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature. In the light of the above statements, choose the correct answer from the options given below :

In a meter bridge experiment to determine the value of unknown resistance, first the resistances $2 \Omega$ and $3 \Omega$ are connected in the left and right gaps of the bridge and the null point is obtained at a distance $l \mathrm{~cm}$ from the left. Now when an unknown resistance $x \Omega$ is connected in parallel to $3 \Omega$ resistance, the null point is shifted by 10 cm to the right of wire. The value of unknown resistance $x$ is $\_\_\_\_$ $\Omega$.

A circular loop of radius $20\text{ cm}$ and resistance $2\ \Omega$ is placed in a time varying magnetic field $\vec{B} = (2t^2 + 2t + 3)\text{ T}$. At $t = 0$, for the plane of the loop being perpendicular to the magnetic field and, the induced current in the loop at $t = 3\text{ s}$ is $\dfrac{\alpha}{50}\text{ A}$. The value of $\alpha$ is _______. (Take $\pi = 22/7$)

A particle of charge $q$ and mass $m$ is projected from origin with an initial velocity $\vec{v} = \left(\dfrac{v_0}{\sqrt{2}}\hat{x} + \dfrac{v_0}{\sqrt{2}}\hat{y}\right)$. There exists a uniform magnetic field $\vec{B} = B_0 \hat{z}$ and a space varying electric field $\vec{E} = E_0 e^{-\lambda x}\hat{x}$ within the region $0 \leq x \leq L$. After travelling a distance such that $x$-coordinate has changed from $x=0$ to $x=L$, the change in the kinetic energy is _______.

In the given circuit below inductance values of $L_1$, $L_2$ and $L_3$ are same. The magnetic energy stored in the entire circuit is $(U_t)$ and that stored in the $L_2$ inductor is $(U_l)$. $U_t / U_l$ is __________. (Ignore the mutual inductance if any) 

A circular loop of radius 7 cm is placed in uniform magnetic field of 0.2 T directed perpendicular to plane of loop. The loop is converted into a square loop in 0.5 s. The EMF induced in the loop is $\_\_\_\_$ mV.

An insulated wire is wound so that it forms a flat coil with $N = 200$ turns. The radius of the innermost turn is $r_1 = 3$ cm, and of the outermost turn $r_2 = 6$ cm. If $20$ mA current flows in it then the magnetic moment will be $\alpha \times 10^{-2}$ A.m$^2$. The value of $\alpha$ is _____.

A parallel plate capacitor is having separation between plates $0.885$ mm. It has a capacitance of $1\,\mu$F when the space between the plates is filled with an insulating material of resistivity $1\times 10^{13}\,\Omega$m and resistance $17.7\times 10^{14}\,\Omega$. Relative permittivity of the insulating material is $\alpha\times 10^7$. The value of $\alpha$ is ________. (Take permittivity of free space $=8.85\times 10^{-12}$ F/m)

 Two identical circular loops $P$ and $Q$ each of radius $r$ are lying in parallel planes such that they have common axis. The current through $P$ and $Q$ are $I$ and $4 I$ respectively in clockwise direction as seen from $O$. The net magnetic field at $O$ is:

Two point charges $2 q$ and $q$ are placed at vertex $A$ and centre of face $C D E F$ of the cube as shown in figure. The electric flux passing through the cube is : 

Six point charges are kept $60^{\circ}$ apart from each other on the circumference of a circle of radius $R$ as shown in figure. The net electric field at the center of the circle is $\_\_\_\_$. ($\epsilon_{0}$ is permittivity of free space) 

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4 R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is : (Take $k=\frac{1}{4 \pi \epsilon_{0}}$ and assume $k Q^{2}>G m^{2}$ where $G$ is the Gravitational constant)

A current carrying circular loop of radius $2$ cm with unit normal $\hat{n} = \dfrac{\hat{k}+\hat{i}}{\sqrt{2}}$ is placed in a magnetic field, $\vec{B} = B_0(3\hat{i}+2\hat{k})$. If $B_0 = 4\times 10^{-3}$ T and current $I=100\sqrt{2}$ A, the torque experienced by the loop is ________ Wb·A. ($\pi=3.14$)

The figure given below shows an $LCR$ series circuit with two switches $S_1$ and $S_2$. When switch $S_1$ is closed keeping $S_2$ open, the phase difference $(\phi)$ between the current and source voltage is $30°$ and phase difference is $60°$ when $S_2$ is closed keeping $S_1$ open. The value of $(3L_1 - L_2)$ is _______ H. 

An infinitely long straight wire carrying current $I$ is bent in a planer shape as shown in the diagram. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular loop is : 

A $5$ mg particle carrying a charge of $5\pi\times 10^{-6}$ C is moving with velocity of $(3\hat{i}+2\hat{k})\times 10^{-2}$ m/s in a region having magnetic field $\vec{B} = 0.1\hat{k}$ Wb/m$^2$. It moves a distance of $\alpha$ meter along $\hat{k}$ when it completes $5$ revolutions. The value of $\alpha$ is ________.

The electrostatic potential in a charged spherical region of radius $r$ varies as $V=a r^{3}+b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a \in_{\mathrm{o}}$. The value of $\alpha$ is $\_\_\_\_$. (permittivity of vacuum is $\epsilon_{0}$)

Two point charges $q_1=3\,\mu C$ and $q_2=-4\,\mu C$ are placed at points $(2\hat{i}+3\hat{j}+3\hat{k})$ and $(\hat{i}+\hat{j}+\hat{k})$ respectively. Force on charge $q_2$ is ________ N. $\left(\text{Take } \dfrac{1}{4\pi\epsilon_0} = 9\times 10^9 \text{ SI Units}\right)$

Two point charges $8 \, \mu C$ and $-2 \, \mu C$ are located at $x = 2$ cm and $x = 4$ cm, respectively on the $x$-axis. The ratio of electric flux due to these charges through two spheres of radii $3$ cm and $5$ cm with their centers at the origin is _______.

Two short electric dipoles $A$ and $B$ having dipole moment $p_1$ and $p_2$ respectively are placed with their axis mutually perpendicular as shown in the figure. The resultant electric field at a point $x$ is making an angle of $60°$ with the line joining points $O$ and $x$. The ratio of the dipole moments $p_2/p_1$ is _______. 

Two point charges of 1 nC and 2 nC are placed at the two corners of equilateral triangle of side 3 cm. The work done in bringing a charge of 3 nC from infinity to the third corner of the triangle is $\_\_\_\_$ $\mu \mathrm{J}$. $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{~N}. \mathrm{m}^{2} / \mathrm{C}^{2}$

Identify the correct statements : A. Electrostatic field lines form closed loops. B. The electric field lines point radially outward when charge is greater than zero. C. The Gauss - Law is valid only for inverse - square force. D. The workdone in moving a charged particle in a static electric field around a closed path is zero. E. The motion of a particle under Coulomb's force must take place in a plane. Choose the correct answer from the options given below :

A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is $\_\_\_\_$. (g: acceleration due to gravity)

Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble. Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is :

Electric field in a region is given by $\vec{E}=A x \hat{i}+B y \hat{j}$, where $A=10 \mathrm{~V} / \mathrm{m}^{2}$ and $B=5 \mathrm{~V} / \mathrm{m}^{2}$. If the electric potential at a point $(10,20)$ is 500 V, then the electric potential at origin is $\_\_\_\_$ V.

Consider a circuit consisting of a capacitor ($20\ \mu$F), resistor ($100\ \Omega$) and two identical diodes as shown in figure. The resistance of diode under forward biasing condition is $10\ \Omega$. The time constant of the circuit is $\alpha \times 10^{-3}$ s. The value of $\alpha$ is _____ 

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed $v$. If $x$ is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to $x^\alpha$, where $\alpha$ is _____.

From the circuit given below, the capacitance between terminals $A$ and $B$ shown in the circuit is ______ $\mu$F. (take $C_1=C_2=C_3=1$ $\mu$F and $C_4=2$ $\mu$F.) 

Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectric ($k_{1}$ and $k_{2}$) in the following fashion. Which of the following is true? $\left(k_{1}>k_{2}\right)$ 

Identify the correct statements : A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination. B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric. C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance. D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces. Choose the correct answer from the options given below :

A parallel plate capacitor has capacitance $C$, when there is vacuum within the parallel plates. A sheet having thickness $\left(\frac{1}{3}\right)^{\mathrm{rd}}$ of the separation between the plates and relative permittivity $K$ is introduced between the plates. The new capacitance of the system is :

A capacitor $P$ with capacitance $10 \times 10^{-6} \mathrm{~F}$ is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor $P$ is connected across another capacitor $Q$ with capacitance $20 \times 10^{-6} \mathrm{~F}$. The charge on capacitor $Q$ when equilibrium is established will be $\alpha \times 10^{-5} C$ (assume capacitor $Q$ does not have any charge initially), the value of $\alpha$ is $\_\_\_\_$。

Two cells of emfs $1$ V and $2$ V and internal resistance $2\,\Omega$ and $1\,\Omega$, respectively connected in parallel, gave a current of $1$ A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be $\dfrac{\alpha}{5}$ A. The value of $\alpha$ is __________.

Two resistors of $200 \, \Omega$ and $400 \, \Omega$ are connected in series with a battery of $100$ V. A bulb rated at $200$ V, $100$ W is connected across the $400 \, \Omega$ resistance. The potential drop across the bulb is _______ V.

The stored charge in the capacitor in steady state of the following circuit is ________ $\mu$C. 

Under steady state condition the potential difference across the capacitor in the circuit is _______ V. 

Refer to the circuit diagram given below. The heat generated across the $6\,\Omega$ resistance in $100$ second is $\dfrac{\alpha}{100}$ J. The value of $\alpha$ is _______. (Nearest integer) 

In an experiment to determine the resistance of a given wire using Ohm's law, the voltmeter and ammeter readings are noted as $10\text{ V}$ and $5\text{ A}$, respectively. The least counts of voltmeter and ammeter are $500\text{ mV}$ and $200\text{ mA}$, respectively. The estimated error in the resistance measurement is _______ $\Omega$.

A voltmeter with internal resistance of $x\ \Omega$ can be used to measure upto $20$ V. In order to increase its measuring range to $30$ V, the required modification is to _____.

Refer to the figure given below, current between terminals $A$ and $B$ is _____ A. 

The voltage and the current between $A$ and $B$ points shown in the circuit are _____. 

Which one of the following is not a measurable quantity?

A Wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances $\left(R_{1}=R_{2}=R_{3}=R_{4}\right)$. When $R_{3}$ resistance is heated to some temperature, its resistance value has gone up by $10 \%$. The potential difference $\left(V_{a}-V_{b}\right)$ (after $R_{3}$ is heated) is $\_\_\_\_$ V. 

In the potentiometer, when the cell in the secondary circuit is shunted with $4 \Omega$ resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell is shunted with $12 \Omega$ resistance, the balance is shifted to a length of 180 cm. The internal resistance of cell is $\_\_\_\_$ $\Omega$

A moving coil galvanometer of resistance $100 \Omega$ shows a full scale deflection for a current of 1 mA. The value of resistance required to convert this galvanometer into an ammeter, showing full scale deflection for a current of 5 mA, is $\_\_\_\_$ $\Omega$

To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm. The least count of scale is 1 cm. The percentage error in the ratio of EMFs is $\_\_\_\_$.

The reading of the ammeter $(A)$ in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is $\_\_\_\_$ A. 

Two resistors of $100 \Omega$ each are connected in series with a 9 V battery. A voltmeter of $400 \Omega$ resistance is connected to measure the voltage drop across one of the resistors. The voltmeter reading is $\_\_\_\_$ V.

A meter bridge with two resistances $R_{1}$ and $R_{2}$ as shown in figure was balanced (null point) at 40 cm from the point $P$. The null point changed to 50 cm from the point $P$, when $16 \Omega$ resistance is connected in parallel to $R_{2}$. The values of resistances $R_{1}$ and $R_{2}$ are $\_\_\_\_$. 

A battery with EMF $E$ and internal resistance $r$ is connected across a resistance $R$. The power consumption in $R$ will be maximum when :

An electric power line having total resistance of $2 \Omega$, delivers 1 kW of power at 250 V. The percentage efficiency of transmission line is $\_\_\_\_$ .

A cylindrical conductor of length 2 m and area of cross-section $0.2 \mathrm{~mm}^{2}$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \mathrm{~m}^{2} / V. s$. The value of $\alpha$ is : (electron concentration $=5 \times 10^{28} / \mathrm{m}^{3}$ and electron charge $\left.=1.6 \times 10^{-19} \mathrm{C}\right)$

Two known resistances of $R \Omega$ and $2 R \Omega$ and one unknown resistance $X \Omega$ are connected in a circuit as shown in the figure. If the equivalent resistance between points $A$ and $B$ in the circuit is $X \Omega$, then the value of $X$ is $\_\_\_\_$ $\Omega$. 

The heat generated in 1 minute between points $A$ and $B$ in the given circuit, when a battery of 9 V with internal resistance of $1 \Omega$ is connected across these points is $\_\_\_\_$ J. 

Two identical small bar magnets each of dipole moment $3\sqrt{5}$ J/T are placed at a center to center separation of $10$ cm, with their axes perpendicular to each other as shown in figure. The value of magnetic field at the point P midway between the magnets is $\alpha \times 10^{-3}$ T. The value of $\alpha$ is ______. ($\mu_0=4\pi \times 10^{-7}$ Tm/A) 

A solenoid has a core made of material with relative permeability $400$. The magnetic field produced in the interior of solenoid is $1.0$ T. The magnetic intensity in SI units is $\alpha \times 10^5$. The value of $\alpha$ is ______. (Free space permeability $\mu_0=4\pi \times 10^{-7}$ SI units.)