Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5 \mathrm {~g/cc}$, the dielectric constant of water will be $\_\_\_\_$. (Take density of water $= 1 \mathrm {~g/cc}$)

The electric field between the two parallel plates of a capacitor of $1.5\mu\mathrm{F}$ capacitance drops to one third of its initial value in $6.6\mu\mathrm{s}$ when the plates are connected by a thin wire. The resistance of this wire is $\_\_\_\_$ $\Omega$. (Given, $\log 3 = 1.1$)

At the centre of a half ring of radius $\mathrm { R } = 10 \mathrm {~cm}$ and linear charge density $4 \mathrm { nCm } ^ { - 1 }$, the potential is $x \pi \mathrm {~V}$. The value of $x$ is $\_\_\_\_$

In the following circuit, the battery has an emf of 2 V and an internal resistance of $\dfrac{2}{3}\,\Omega$. The power consumption in the entire circuit is $\_\_\_\_$ W.

A wire of resistance $R$ and radius $r$ is stretched till its radius became $r / 2$. If new resistance of the stretched wire is $x R$, then value of $x$ is $\_\_\_\_$

The current in a conductor is expressed as $I = 3t ^ { 2 } + 4t ^ { 3 }$, where $I$ is in Ampere and $t$ is in second. The amount of electric charge that flows through a section of the conductor during $t = 1 \mathrm {~s}$ to $t = 2 \mathrm {~s}$ is $\_\_\_\_$ C.

The current flowing through the $1 \Omega$ resistor is $\frac { n } { 10 } \mathrm {~A}$. The value of $n$ is $\_\_\_\_$
[Figure]

Two circular coils $P$ and $Q$ of 100 turns each have same radius of $\pi$ cm. The currents in $P$ and $Q$ are 1 A and 2 A respectively. $P$ and $Q$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is $\sqrt{x}$ mT, where $x =$ $\_\_\_\_$. [Use $\mu_0 = 4\pi \times 10^{-7}$ T m A$^{-1}$]

A regular polygon of 6 sides is formed by bending a wire of length $4\pi$ meter. If an electric current of $4\pi\sqrt{3}$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $x \times 10 ^ { - 7 } \mathrm {~T}$. The value of $x$ is $\_\_\_\_$.

The magnetic field existing in a region is given by $\vec { B } = 0.2 ( 1 + 2 x ) \hat { k } \mathrm {~T}$. A square loop of edge 50 cm carrying 0.5 A current is placed in $x - y$ plane with its edges parallel to the $x - y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is $\_\_\_\_$ mN.

A 2 A current carrying straight metal wire of resistance $1\Omega$, resistivity $2 \times 10^{-6}\Omega\mathrm{m}$, area of cross-section $10\mathrm{~mm}^2$ and mass 500 g is suspended horizontally in mid air by applying a uniform magnetic field $\vec{B}$. The magnitude of $B$ is $\_\_\_\_$ $\times 10^{-1}\mathrm{~T}$ (given, $\mathrm{g} = 10\mathrm{~m/s}^2$).

A square loop of edge length 2 m carrying current of 2 A is placed with its edges parallel to the $x y$ axis. A magnetic field is passing through the $x - y$ plane and expressed as $\vec { B } = B _ { 0 } ( 1 + 4 x ) \hat { k }$, where $B _ { 0 } = 5 \mathrm {~T}$. The net magnetic force experienced by the loop is $\_\_\_\_$ N.

The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8\,\Omega$ varies with time (in seconds) as $\phi = 5t^2 - 36t + 1$. The induced current in the circuit at $t = 2$ s is $\_\_\_\_$ A.

A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the $x$-axis and $y$-axis respectively moves with a velocity of $5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ in the positive $x$ axis direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10 ^ { - 3 } \mathrm {~T} \mathrm {~cm} ^ { - 1 }$ along the negative $x$ direction and it is decreasing with time at the rate of $10 ^ { - 3 } \mathrm {~T} \mathrm {~s} ^ { - 1 }$. If the resistance of the loop is $6 \mathrm {~m\Omega}$, the power dissipated by the loop as heat is $\_\_\_\_$ $\times 10 ^ { - 9 } \mathrm {~W}$.

An alternating current at any instant is given by $i = [ 6 + \sqrt { 56 } \sin ( 100 \pi t + \pi / 3 ) ]$ A. The $rms$ value of the current is $\_\_\_\_$ A.

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $20\mu\mathrm{F}$ is $\_\_\_\_$ V. $$\mathrm{V} = 50\sqrt{2}\sin 100t \text{ volt}$$

When a coil is connected across a 20 V dc supply, it draws a current of 5 A. When it is connected across $20 \mathrm {~V} , 50 \mathrm {~Hz}$ ac supply, it draws a current of 4 A. The self inductance of the coil is $\_\_\_\_$ mH. (Take $\pi = 3$)

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm . The focal length of the lens used is $\_\_\_\_$ cm.

Light from a point source in air falls on a convex curved surface of radius 20 cm and refractive index 1.5. If the source is located at 100 cm from the convex surface, the image will be formed at $\_\_\_\_$ cm from the object.

The distance between object and its 3 times magnified virtual image as produced by a convex lens is 20 cm. The focal length of the lens used is $\_\_\_\_$ cm.

Two wavelengths $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are used in Young's double slit experiment. $\lambda _ { 1 } = 450 \mathrm {~nm}$ and $\lambda _ { 2 } = 650 \mathrm {~nm}$. The minimum order of fringe produced by $\lambda _ { 2 }$ which overlaps with the fringe produced by $\lambda _ { 1 }$ is $n$. The value of $n$ is $\_\_\_\_$.

In Young's double slit experiment, carried out with light of wavelength $5000\,\text{\AA}$, the distance between the slits is 0.3 mm and the screen is at 200 cm from the slits. The central maximum is at $x = 0\mathrm{~cm}$. The value of $x$ for third maxima is $\_\_\_\_$ mm.

In a Young's double slit experiment, the intensity at a point is $\left( \frac { 1 } { 4 } \right) ^ { \text {th} }$ of the maximum intensity, the minimum distance of the point from the central maximum is $\_\_\_\_$ $\mu \mathrm { m }$. (Given: $\lambda = 600 \mathrm {~nm} , \mathrm {~d} = 1.0 \mathrm {~mm} , \mathrm { D } = 1.0 \mathrm {~m}$)

A nucleus has mass number $A_1$ and volume $V_1$. Another nucleus has mass number $A_2$ and volume $V_2$. If relation between mass number is $A_2 = 4A_1$, then $\dfrac{V_2}{V_1} =$ $\_\_\_\_$.

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is $\frac { 1000 } { x }$, where $x$ is $\_\_\_\_$.