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A non-relativistic proton beam passes without deviation through the region of space where uniform transverse mutually perpendicular electric and magnetic fields of $E=120{\text{kVm}}^{-1}$ and $B=50\text{mT}$. Then the beam strikes a grounded target. Find the force with which the beam hits on the target if the beam current is equal to $I=0.80\text{mA}$.[Take mass of proton and its charge as $m_e=1.67\times {10}^{-27}\text{kg}$ and $q=1.6\times {10}^{-19}\text{C}$ respectively.]

A gas undergoes a cyclic process $a\to b\to c\to a$ which is as shown in the $PV$ diagram. The process $a\to b$ is isothermal , $b\to c$ is adiabatic and $c\to a$ is a straight line on the $P-V$ diagram. Work done in process $ab$ and $bc$ is $5\text{J}$ and $4\text{J}$ respectively. Calculate the efficiency of the cycle, if the area enclosed by the diagram $\text{abca}$ in the figure is $3\text{J}$.

Points I, II and III in the following plot respectively correspond to $u_{mp}$ (most probable velocity).



Choose the correct order in increasing order of their most probable velocity.

A sample of mono-atomic ideal gas is taken through a cyclic process ABCDA. A graph is plotted in which heat exchanged by the gas upto an instant is shown on x - axis and work done by the gas upto that instant is shown on y - axis.





The given cyclic process can be represented on a pressure - volume graph as

An ideal refrigerator has a freezer at a temperature of $-{13}^{\circ }\text{C}$. The coefficient of performance of the engine used is $5$. The temperature of the air (to which heat is rejected) will be

The coil having one turn is made into coil of two turns. If the same current is passed in both the cases, then the ratio of the magnetic induction at their centers will be

The angle of twist of a closely helical spring under an axial torque is given by :

Energy of photon whose frequency is ${10}^{12}MHz$, will be

If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, then which of the following options is correct?

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the y-axis in the y-z plane at $z=\frac{\sqrt{3}}{2}a$. If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is $\frac{\lambda L}{n{\epsilon }_0}$ (${\epsilon }_0=$ permittivity of free space), then the value of $n$ is

Figure shows three forces applied to a block that moves leftwards by $3\text{m}$ over a smooth floor. The magnitudes of forces are $F_1=3\text{N},F_2=9\text{N}$, and $F_3=5\text{N}$. The net work done on the block by the three forces is

Derivation and condition for coherent and incoherent waves addition.

Consider the figure shown below. If all the surfaces are smooth except that between mass $m$ and $2m$ then the acceleration (in ${\text{m/s}}^2$) of the mass $m$ is

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:

(a) Show p_i = p’_i + m_iV

Where p_i is the momentum of the i^th particle (of mass m_i) and p′ _i = m_i v′ _i. Note v′ _i is the velocity of the i^th particle relative to the centre of mass.

Also, prove using the definition of the centre of mass

(b) Show K = K′ + ½MV²

Where K is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and MV²/2 is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.

(c) Show L = L′ + R × MV

Where is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember r’_i = r_i – R; rest of the notation is the standard notation used in the chapter. Note L′ and MR × V can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.

(d) Show

Further, show that

where τ′_ext is the sum of all external torques acting on the system about the centre of mass.

(Hint: Use the definition of centre of mass and Newton’s Third Law. Assume the internal forces between any two particles act along the line joining the particles.)

The average kinetic energy of a gas molecule at ${27}^{\circ }$ is $6.21\times {10}^{-21}J.$ The average kinetic energy at ${227}^{\circ }$ will be

Two concentric spherical shells have masses $M_1,M_2$ and radii, $R_1,R_2(R_1<R_2).$ What is the force exerted by this system on a particle of mass $m_1$ if it is placed at a distance $\frac{(R_1+R_2)}{2}$ from the centre?

A string of length $l$ along $x-$ axis is fixed at both ends and is vibrating in second harmonic. If amplitude of incident wave is $2.5\text{mm}$, the equation of standing wave is (T is tension and $\mu$ is linear density)

A light ray is incident on a transparent sphere of index $=\sqrt{2}$ , at an angle of incidence $={45}^{\circ }$. What is the deviation of a tiny fraction of the ray, which enters the sphere, undergoes two internal reflections, and then refracts out into air?

Two blocks $\text{A}$ and $\text{B}$ of equal mass $m=1.0$$\text{kg}$ are lying on a smooth horizontal surface as shown in figure. A spring of force constant $k=200\text{N/m}$ is fixed at one end of block $\text{A}$. Block $\text{B}$ collides with block $\text{A}$ with velocity $v_0=2.0$$\text{m/s}$. Find the maximum compression of the spring.

Given the speed-time graph of a golf ball shot from ground at t = 0.

What will be the time of flight?

A photon moves vertically up in a region with gravitational field g downwards. The frequency of photon at an instant is ν .After it has moved up by a small height h,

How is the magnetic field due to a straight current carrying wire affected if current in the wire is (a) decreased, (b) reversed?

An open lift is moving upwards with velocity 10 m/s. It has an upward acceleration of 2 m/s${}^2$. A ball is projected upwards with velocity 20 m/s relative to ground. Find the time when ball again meets the lift.

A solid sphere of radius ${}^{\prime }{r}^{\prime }$ and mass ${}^{\prime }{m}^{\prime }$ is rotating about an axis passing through its COM with an angular speed ${}^{\prime }{\omega }^{\prime }$ and has rotational kinetic energy $k_1$. If the radius of the sphere is halved and angular speed is doubled, then its rotational kinetic energy becomes $k_2$. Which of the following options is correct ?

In a chamber, a uniform
magnetic field of 6.5 G (1 G = 10⁻⁴ T) is
maintained. An electron is shot into the field with a speed of 4.8 ×
10⁶ m s⁻¹ normal to the field. Explain
why the path of the electron is a circle. Determine the radius of the
circular orbit. (e = 1.6 × 10⁻¹⁹ C, m_e= 9.1×10⁻³¹ kg)

An element dl=dx icap is placed at the origin and carries a large current I =10A .What is the magnetic field on the y-axis at a distance of 0.5m .dx =1cm?

The motor of an engine is rotating about its axis at an angular speed of $120\text{rpm}$. It comes to rest in $10\text{s}$, after being swiched off. Assuming constant acceleration, the number of revolutions made by it before coming to rest are