Four charges equal to $- Q$ are placed at the four corners of a square and a charge $q$ is at its centre. If the system is in equilibrium the value of $q$ is
(1) $- \frac { Q } { 4 } ( 1 + 2 \sqrt { 2 } )$
(2) $\frac { Q } { 4 } ( 1 + 2 \sqrt { 2 } )$
(3) $- \frac { Q } { 2 } ( 1 + 2 \sqrt { 2 } )$
(4) $\frac { Q } { 2 } ( 1 + 2 \sqrt { 2 } )$

A charged oil drop is suspended in a uniform field of $3 \times 10 ^ { 4 } \mathrm {~V} / \mathrm { m }$ so that it neither falls nor rises. The charge on the drop will be (take the mass of the charge $= 9.9 \times 10 ^ { - 15 } \mathrm {~kg}$ and $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ )
(1) $3.3 \times 10 ^ { - 18 } \mathrm { C }$
(2) $3.2 \times 10 ^ { - 18 } \mathrm { C }$
(3) $1.6 \times 10 ^ { - 18 } \mathrm { C }$
(4) $4.8 \times 10 ^ { - 18 } \mathrm { C }$

The resistance of the series combination of two resistances is $S$. When they are joined in parallel the total resistance is $P$. If $S = n P$, then the minimum possible value of $n$ is
(1) 4
(2) 3
(3) 2
(4) 1

An electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the length and radii of the wires are in the ratio of $4/3$ and $2/3$, then the ratio of the currents passing through the wire will be
(1) 3
(2) $1 / 3$
(3) $8 / 9$
(4) 2

In a metre bridge experiment null point is obtained at 20 cm from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then where will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?
(1) 50 cm
(2) 80 cm
(3) 40 cm
(4) 70 cm

The electrochemical equivalent of a metal is $3.3 \times 10 ^ { - 7 } \mathrm {~kg}$ per coulomb. The mass of the metal liberated at the cathode when a 3 A current is passed for 2 seconds will be
(1) $19.8 \times 10 ^ { - 7 } \mathrm {~kg}$
(2) $9.9 \times 10 ^ { - 7 } \mathrm {~kg}$
(3) $6.6 \times 10 ^ { - 7 } \mathrm {~kg}$
(4) $1.1 \times 10 ^ { - 7 } \mathrm {~kg}$

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is 2 s. The magnet is cut along its length into three equal parts and three parts are then placed on each other with their like poles together. The time period of this combination will be
(1) 2 s
(2) $2 / 3 \mathrm {~s}$
(3) $2 \sqrt { 3 } \mathrm {~s}$
(4) $2 / \sqrt { 3 } \mathrm {~s}$

A long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is $B$. It is then bent into a circular loop of $n$ turns. The magnetic field at the centre of the coil will be
(1) $nB$
(2) $n ^ { 2 } B$
(3) $2 n B$
(4) $2 n ^ { 2 } B$

The magnetic field due to a current carrying circular loop of radius 3 cm at a point on the axis at a distance of 4 cm from the centre is $54 \mu \mathrm {~T}$. What will be its value at the centre of the loop?
(1) $250 \mu \mathrm {~T}$
(2) $150 \mu \mathrm {~T}$
(3) $125 \mu \mathrm {~T}$
(4) $75 \mu \mathrm {~T}$

Two long conductors, separated by a distance $d$ carry current $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 2 }$ in the same direction. They exert a force $F$ on each other. Now the current in one of them is increased to two times and its direction reversed. The distance is also increased to $3d$. The new value of the force between them is
(1) $-2F$
(2) $F / 3$
(3) $-2F / 3$
(4) $-F / 3$

A coil having $n$ turns and resistance $4R\,\Omega$ is connected with a resistance $R$. This combination is moved in time $t$ seconds from a magnetic field $W _ { 1 }$ weber to $W _ { 2 }$ weber. The induced current in the circuit is
(1) $- \frac { W _ { 2 } - W _ { 1 } } { 5 R n t }$
(2) $- \frac { \left( \mathrm { W } _ { 2 } - \mathrm { W } _ { 1 } \right) } { 5 \mathrm { Rt } }$
(3) $- \frac { W _ { 2 } - W _ { 1 } } { R n t }$
(4) $- \frac { \mathrm { n } \left( \mathrm { W } _ { 2 } - \mathrm { W } _ { 1 } \right) } { \mathrm { Rt } }$

In a uniform magnetic field of induction $B$ a wire in the form of a semicircle of radius $r$ rotates about the diameter of the circle with angular frequency $\omega$. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is $R$, the mean power generated per period of rotation is
(1) $\frac { B \pi r ^ { 2 } \omega } { 2 R }$
(2) $\frac { \left( B \pi r ^ { 2 } \omega \right) ^ { 2 } } { 2 R }$
(3) $\frac { ( \mathrm { B } \pi \mathrm { r } \omega ) ^ { 2 } } { 2 \mathrm { R } }$
(4) $\frac { \left( B \pi r ^ { 2 } \omega \right) ^ { 2 } } { 8 R }$

In an LCR series a.c. circuit, the voltage across each of the components, $L$, $C$ and $R$ is 50 V. The voltage across the LC combination will be
(1) 50 V
(2) $50 \sqrt { 2 } \mathrm { V }$
(3) 100 V
(4) 0 V (zero)

In a LCR circuit capacitance is changed from $C$ to $2C$. For the resonant frequency to remain unchanged, the inductance should be changed from $L$ to
(1) 4L
(2) 2L
(3) $L / 2$
(4) $L / 4$

A plane convex lens of refractive index 1.5 and radius of curvature 30 cm is silvered at the curved surface. Now this lens has been used to form the image of an object. At what distance from this lens should an object be placed in order to have a real image of the size of the object?
(1) 20 cm
(2) 30 cm
(3) 60 cm
(4) 80 cm

The angle of incidence at which reflected light is totally polarized for reflection from air to glass (refractive index $n$), is
(1) $\sin ^ { - 1 } ( n )$
(2) $\sin ^ { - 1 } ( 1 / n )$
(3) $\tan ^ { - 1 } ( 1 / n )$
(4) $\tan ^ { - 1 } ( n )$

The maximum number of possible interference maxima for slit-separation equal to twice the wavelength in Young's double-slit experiment is
(1) infinite
(2) five
(3) three
(4) zero

A radiation of energy $E$ falls normally on a perfectly reflecting surface. The momentum transferred to the surface is
(1) $E / c$
(2) $2 \mathrm { E } / \mathrm { c }$
(3) $Ec$
(4) $E / c ^ { 2 }$

According to Einstein's photoelectric equation, the plot of the kinetic energy of the emitted photo electrons from a metal vs the frequency of the incident radiation gives a straight line whose slope
(1) depends on the nature of the metal used
(2) depends on the intensity of the radiation
(3) depends both on the intensity of the radiation and the metal used
(4) is the same for all metals and independent of the intensity of the radiation.

The work function of a substance is 4.0 eV. What is the longest wavelength of light that can cause photoelectron emission from this substance?

Average density of the earth
(1) does not depend on $g$
(2) is a complex function of g
(3) is directly proportional to g
(4) is inversely proportional to g

The change in the value of $g$ at a height '$h$' above the surface of the earth is the same as at a depth '$d$' below the surface of earth. When both '$d$' and '$h$' are much smaller than the radius of earth, then which one of the following is correct?
(1) $d = \frac{h}{2}$
(2) $\mathrm{d} = \frac{3\mathrm{h}}{2}$
(3) $d = 2\mathrm{h}$
(4) $d = h$

A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere (you may take $\mathrm{G} = 6.67 \times 10^{-11} \mathrm{Nm}^2/\mathrm{kg}^2$)
(1) $13.34 \times 10^{-10} \mathrm{~J}$
(2) $3.33 \times 10^{-10} \mathrm{~J}$
(3) $6.67 \times 10^{-9} \mathrm{~J}$
(4) $6.67 \times 10^{-10} \mathrm{~J}$

A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is $4/3$ and the fish is 12 cm below the surface, the radius of this circle in cm is
(1) $36\sqrt{7}$
(2) $36/\sqrt{7}$
(3) $36\sqrt{5}$
(4) $4\sqrt{5}$

A thin glass (refractive index 1.5) lens has optical power of $-5\,D$ in air. Its optical power in a liquid medium with refractive index 1.6 will be
(1) 1 D
(2) $-$1 D
(3) 25 D
(4) None of these