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A short magnetic dipole is placed at the origin with its dipole moment directed along the $+vex$- axis . If magnetic induction at a point $\text{P}(r,0)$ is $B\hat{i}$, the magnetic induction at point $\text{Q}(0,2r)$ will be:

In a capacitor of capacitance $20\mu \text{F}$ , the distance between the plates is $2\text{mm}$. If a dielectric slab of width $1\text{mm}$ and dielectric constant $2$ is inserted between the plates, then the new capacitance will be :

Metals have resistivities of the order of

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes $2cm,1m{s}^{-1}and10m{s}^{-2}$ at a certain instant. Find the amplitude and the time period of the motion.

An electromagnetic wave going through vacuum is described by $E=E_0\sin (kx-\omega t)$. Which of the following is/are independent of the wavelength?

Find the number of free electrons per unit volume of a metallic wire of density ${10}^4{\text{kg/m}}^3$ and atomic mass number $100$. The number of free electrons per atom is one.

A ${20}^{\circ }$ full depth involute spur pinion of 4 mm module and 21 teeth is to transmit 15 kW at 960 rpm. Its face width is 25 mm.

The tangential force transmitted (in N) is

Find the position vector of the centre of mass ${\overrightarrow{r}}_{\text{com}}$ of an asymmetric uniform bar of negligible area of cross-section, shown in the figure given below.

Determine height $h$ of oil in the U tube as shown in figure. Density of oil $=0.9\text{g/cc}$; density of liquid is $1.6\text{g/cc}$ and density of mercury $=13.6\text{g/cc}$.

For the given uniformly charged ring, magnitude of net electric field at point $\text{P}$ is

Two satellites $\text{A}$ and $\text{B}$, have ratio of masses $3:1$ are in circular orbits of radii $r$ and $4r$. Then ratio of total mechanical energy of $\text{A}$ and $\text{B}$ is

A block of mass $15\text{kg}$ is acted upon by a force $25\text{N}$ at an angle ${37}^{\circ }$ with the horizontal in a direction as shown in figure. The coefficient of static friction($\mu$) between the block and horizontal surface is $0.2$ . Find the frictional force acting on the block, so that it doesn't move. Take $(g=10{\text{m/s}}^2)$.

Across a conductor of length $40\text{cm}$, a potential difference of $10\text{V}$ is maintained. The mobility of electrons if the drift velocity of electrons is $5\times {10}^{-6}\text{m/s}$ is

Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections $A,B$ and $C$ are shown. Section $B$ and $C$ have equal areas of cross- section. If $P_A,P_B$ and $P_C$ are the pressures at $A,B$ and $C$ respectively, then

A 50 m long train passes over a bridge at a speed of 30 km/h, if it takes 36 secs cross the bridge, what is the length of the bridge?

A sinusoidal wave represented by $y_i=0.3\cos (2x-40t)$ is travelling along a string towards a fixed end at $x=0$. Then, the equation of the reflected wave will be:-
(Assume there is no loss of energy at the boundary)

A wire carrying current $I$ has the shape as shown in adjoining figure. Linear parts of the wire are very long and parallel to $X$-axis while semicircular portion of radius $R$ is lying in $Y-Z$ plane. Magnetic field at point $O$ is

A block of mass $8\text{kg}$, moving in x-direction with a constant speed of $15\text{m/s}$, is subjected to a retarding force of $F=10x\text{N}$ during its travel from $x=10\text{m}$ to $x=15\text{m}$. Find its final kinetic energy.

In the circuit diagram shown below :

The $\text{SI}$ unit of density is

An air blower handles $\dot{m}$ J kg/s of air at $T_c$ and consumes a power of $P_0$ kW. The air enters the blower at $V_c$ m/s and exits at double the velocity at inlet. Considering adiabatic conditions. the exit temperature of air is

The focal length of the objective and eyepiece of a microscope are $2\text{cm}$ and $5\text{cm}$ respectively, and the distance between them is $20\text{cm}$. Find the distance of the object from the objective, when the final image seen by the eye is $25\text{cm}$ from the eyepiece.

A stone of mass $m$ is thrown with a velocity $u$ at a block of mass $M$ which is initially at rest. The collision between the bodies is elastic. If the final velocities of the stone and the block are in the ratio $-4:1$, then what is the ratio of the mass of the stone to the mass of the block?

A toy aeroplane of mass $m_a=3\text{kg}$ is moving with a velocity of $10\text{m/s}$ in a horizontal direction. If a bird of mass $m_b=2\text{kg}$ sits on the toy aeroplane, find the final velocity of (toy+bird) system.(Initial velocity of the bird is zero.)

Two similar springs $P$ and $Q$ have spring constants $k_P$ and $k_Q$, such that $k_P>k_Q$. They are stretched, first by the same amount (case a ), then by the same force (case b ). The work done by the springs $W_P$ and $W_Q$ are related as, in case (a) and case (b), respectively

Force $F$ on a particle moving in a straight line varies with distance $d$ as shown in the figure

The work done on the particle during its displacement of $12\text{m}$ is

(a) In young’s double slit experiment, deduce the conditions for obtaining constructive and destructive interference fringes. Hence, deduce the expression for the fringe width.

(b)Show that the fringe pattern on the screen is actually a superposition of slit diffraction from each slit.

(c) What should be the width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern, for green light of wavelength 500 nm, if the separation between two slits is 1 mm?

OR

(a) Two thin convex lenses $L_1$and $L_2$ of focal lengths $f_1$and$f_2$ respectively, are placed coaxially in contact. An object is placed at a point beyond the focus of lens $L_1$ . Draw a ray diagram to show the image formation by the combination and hence derive the expression for the focal length of the combined system.

(b) A ray PQ incident on the face AB of a prism ABC, as shown in the figure, emerges from the face AC such that AQ = AR

Draw the ray diagram showing the passage of the ray through the prism. If the angle of the prism is ${60}^0$ and refractive index of the material of prism is $\sqrt{3}$. Determine the angle of incidence and angle of deviation

I have an insulator with dielectric constant k. If I place it in a region in vacuum with a field E, what is the induced field inside the dielectric?

A disc has mass $9\text{m}$. A hole of radius $\frac{R}{3}$ is cut from it as shown in the figure. The moment of inertia of remaining part about an axis passing through the centre ‘$O$’ of the disc and perpendicular to the plane of the disc is:

1. 0.18

Two small spheres with mass $m_1$ and $m_2$ hang from massless insulating threads of length $l_1$ and $l_2$. The two spheres carry charges $q_1$ and $q_2$ respectively. The spheres hang such that they are on the same horizontal level and the threads are inclined to the vertical at angles ${\theta }_1$ and ${\theta }_2$. Which of the conditions is required if ${\theta }_1={\theta }_2$?

A
spherical capacitor has an inner sphere of radius 12 cm and an outer
sphere of radius 13 cm. The outer sphere is earthed and the inner
sphere is given a charge of 2.5 µC. The space between the
concentric spheres is filled with a liquid of dielectric constant 32.

(a) Determine
the capacitance of the capacitor.

(b) What
is the potential of the inner sphere?

(c) Compare
the capacitance of this capacitor with that of an isolated sphere of
radius 12 cm. Explain why the latter is much smaller.