The box of a pin hole camera, of length $$L,$$ has a hole of radius a. It is assumed that when the hole is illuminated by a parallel beam of light of wavelength $$\lambda$$ the spread of the spot (obtained on the opposite wall of the camera) is the sum of its geometrical spread and the spread due to diffraction. The spot would then have its minimum size (say $$b_{\min }$$) when :

Two stars are 10 light years away from the earth. They are seen through a telescope of objective diameter 30 cm. The wavelength of light is 600 nm. To see the stars just resolved by the telescope, the minimum distance between them should be (1 light year = 9.46 $$\times$$ 1015 m) of the order of :

In Young’s double slit experiment, the distance between slits and the screen is 1.0 m and monochromatic light of 600 nm is being used. A person standing near the slits is looking at the fringe pattern. When the separation between the slits is varied, the interference pattern disappears for a particular distance d0 between the slits. If the angular resolution of the eye is ({{{1}} \over {60}})^o, the value of d0 is close to :

A combination of capacitors is set up as shown in the figure. The magnitude of the electric field, due to a point charge $$Q$$ (having a charge equal to the sum of the charges on the $$4$$ $$\mu F$$ and $$9$$ $$\mu F$$ capacitors), at a point distance $$30$$ $$m$$ from it, would equal :

Figure shows a network of capacitors where the numbers indicates capacitances in micro Farad. The value of capacitance C if the equivalent capacitance between point A and B is to be 1 $$\mu$$F is :

Three capacitors each of 4 $$\mu$$F are to be connected in such a way that the effective capacitance is 6 $$\mu$$F. This can be done by connecting them :

A pipe open at both ends has a fundamental frequency $$f$$ in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now :

A toy-car, blowing its horn, is moving with a steady speed of 5 m/s, away from a wall. An observer, towards whom the toy car is moving, is able to hear 5 beats per second. If the velocity of sound in air is 340 m/s, the frequency of the horn of the toy car is close to :

Two engines pass each other moving in opposite directions with uniform speed of 30 m/s. One of them is blowing a whistle of frequency 540 Hz. Calculate the frequency heard by driver of second engine before they pass each other. Speed of sound is 330 m/sec :

A uniform string of length $$20$$ $$m$$ is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the supports is : (take $$g=10m{s}^{-2}$$ )

Two identical wires $$A$$ and $$B,$$ each of length $${}^{\prime }{l}^{\prime }$$, 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 $${}^{\prime }{a}^{\prime }$$. If $$B_A$$ and $$B_B$$ are the values of magnetic fields at the centres of the circle and square respectively, then the ratio {{{B_A}} \over {{B_B}}} is:

To know the resistance G of a galvanometer by half deflection method, a battery of emf VE and resistance R is used to deflect the galvanometer by angle $$\theta$$. If a shunt of resistance S is needed to get half deflection then G, R and S are related by the equation :

A magnetic dipole is acted upon by two magnetic fields which are inclined to each other at an angle of 75o. One of the fields has a magnitude of 15 mT. The dipole attains stable equilibrium at an angle of 30o with this field. The magnitude of the other field (in mT ) is close to

A particle of mass m is acted upon by a force F given by the empirical law F ={R \over {{t^2}}}\,v\left( t \right). If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot :

A rocket is fired vertically from the earth with an acceleration of 2g, where g is the gravitational acceleration. On an inclined plane inside the rocket, making an angle $$\theta$$ with the horizontal, a point object of mass m is kept. The minimum coefficient of friction $$\mu$$min between the mass and the inclined surface such that the mass does not move is :

Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed ‘v’ in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities $$\sigma$$1 and $$\sigma$$2 are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects) :

A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as B = B0e{^{{{ - t} \over r}}} , where B0 and $$\tau$$ are constants, at time t = 0. If the resistance of the loop is R then the heat generated in the loop after a long time (t $$\to$$ $$\infty$$) is :

A roller is made by joining together two cones at their vertices $$0$$. It is kept on two rails $$AB$$ and $$CD$$, which are placed asymmetrically (see figure), with its axis perpendicular to $$CD$$ and its center $$O$$ at the center of line joining $$AB$$ and $$CD$$ (see figure). It is given a light push so that it starts rolling with its center $$O$$ moving parallel to $$CD$$ in the direction shown. As it moves, the roller will tend to :

A particle of mass m is moving along the side of a square of side ‘a’, with a uniform speed v in the x-y plane as shown in the figure : Which of the following statements is false for the angular momentum $$\overrightarrow{L}$$ about the origin ?

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to : (Take the radius of the drum to be 1.25 m and its axle to be horizontal) :