Consider a block of conducting material of resistivity $${}^{\prime }{\rho }^{\prime }$$ shown in the figure. Current $${}^{\prime }{I}^{\prime }$$ enters at $${}^{\prime }{A}^{\prime }$$ and leaves from $${}^{\prime }{D}^{\prime }$$. We apply superposition principle to find voltage $${}^{\prime }\Delta V^{\prime }$$ developed between $${}^{\prime }{B}^{\prime }$$ and $${}^{\prime }{C}^{\prime }$$. The calculation is done in the following steps: (i) Take current $${}^{\prime }{I}^{\prime }$$ entering from $${}^{\prime }{A}^{\prime }$$ and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $$E(r)$$ at distance $${}^{\prime }{r}^{\prime }$$ from A by using Ohm's law $$E=\rho j,$$ where $$j$$ is the current per unit area at $${}^{\prime }{r}^{\prime }$$. (iii) From the $${}^{\prime }{r}^{\prime }$$ dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$. (iv) Repeat (i), (ii) and (iii) for current $${}^{\prime }{I}^{\prime }$$ leaving $${}^{\prime }{D}^{\prime }$$ and superpose results for $${}^{\prime }{A}^{\prime }$$ and $${}^{\prime }{D}^{\prime }.$$ For current entering at $$A,$$ the electric field at a distance $${}^{\prime }{r}^{\prime }$$ from $$A$$ is

Consider a block of conducting material of resistivity $${}^{\prime }{\rho }^{\prime }$$ shown in the figure. Current $${}^{\prime }{I}^{\prime }$$ enters at $${}^{\prime }{A}^{\prime }$$ and leaves from $${}^{\prime }{D}^{\prime }$$. We apply superposition principle to find voltage $${}^{\prime }\Delta V^{\prime }$$ developed between $${}^{\prime }{B}^{\prime }$$ and $${}^{\prime }{C}^{\prime }$$. The calculation is done in the following steps: (i) Take current $${}^{\prime }{I}^{\prime }$$ entering from $${}^{\prime }{A}^{\prime }$$ and assume it to spread over a hemispherical surface in the block. (ii) Calculate field $$E(r)$$ at distance $${}^{\prime }{r}^{\prime }$$ from A by using Ohm's law $$E=\rho j,$$ where $$j$$ is the current per unit area at $${}^{\prime }{r}^{\prime }$$. (iii) From the $${}^{\prime }{r}^{\prime }$$ dependence of $$E(r)$$, obtain the potential $$V(r)$$ at $$r$$. (iv) Repeat (i), (ii) and (iii) for current $${}^{\prime }{I}^{\prime }$$ leaving $${}^{\prime }{D}^{\prime }$$ and superpose results for $${}^{\prime }{A}^{\prime }$$ and $${}^{\prime }{D}^{\prime }.$$ $$\Delta V$$ measured between $$B$$ and $$C$$ is

A $$5V$$ battery with internal resistance $$2\Omega$$ and a $$2V$$ battery with internal resistance $$1\Omega$$ are connected to a $$10\Omega$$ resistor as shown in the figure. The current in the $$10\Omega$$ resistor is

A block of mass $$0.50$$ $$kg$$ is moving with a speed of $$2.00$$ $$m{s}^{-1}$$ on a smooth surface. It strike another mass of $$1.0$$ $$kg$$ and then they move together as a single body. The energy loss during the collision is :

A thin rod of length $${}^{\prime }{L}^{\prime }$$ is lying along the $$x$$-axis with its ends at $$x=0$$ and $$x=L$$. Its linear density (mass/length) varies with $$x$$ as k{\left( {{x \over L}} \right)^n}, where $$n$$ can be zero or any positive number. If the position $$X_{CM}$$ of the center of mass of the rod is plotted against $${}^{\prime }{n}^{\prime },$$ which of the following graphs best approximates the dependence of $$X_{CM}$$ on $$n$$?

A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in a soap-water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?

A jar is filled with two non-mixing liquids $$1$$ and $$2$$ having densities $${\rho }_1$$ and $${\rho }_2$$ respectively. A solid ball, made of a material of density $${\rho }_3$$, is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $${\rho }_1$$ , $${\rho }_1$$ and $${\rho }_3$$ ?

A spherical solid ball of volume $$V$$ is made of a material of density $${\rho }_1$$. It is falling through a liquid of density $${\rho }_2\left({\rho }_{2}0\right).$$ The terminal speed of the ball is

A working transistor with its three legs marked $$P,Q$$ and $$R$$ is tested using a multi-meter. No conduction is found between $$P$$ and $$Q$$. By connecting the common (negative) terminal of the multi-meter to $$R$$ and the other (positive) terminal to $$P$$ or $$Q,$$ some resistance is seen on the multi-meter. Which of the following is true for the transistor?

In the circuit below, $$A$$ and $$B$$ represent two inputs and $$C$$ represents the output. The circuit represents

An athlete in the olympic games covers a distance of $$100$$ $$m$$ in $$10$$ $$s.$$ His kinetic energy can be estimated to be in the range

This question contains Statement- 1 and Statement- 2. Of the four choices given after the statements, choose the one that best describes the two statements: Statement- 1: Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion and Statement- 2: For heavy nuclei, binding energy per nucleon increases with increasing $$Z$$ while for light nuclei it decreases with increasing $$Z.$$

Suppose an electron is attracted towards the origin by a force {k \over r} where $${}^{\prime }{k}^{\prime }$$ is a constant and $${}^{\prime }{r}^{\prime }$$ is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the $$n^{th}$$ orbital of the electron is found to be $${}^{\prime }{r_n}^{\prime }$$ and the kinetic energy of the electron to be $${}^{\prime }{T_n}^{\prime }.$$ Then which of the following is true?

The speed of sound in oxygen $$\left(O_2\right)$$ at a certain temperature is $$460m{s}^{-1}.$$ The speed of sound in helium $$(He)$$ at the same temperature will be (assume both gases to be ideal)

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $$V_1$$ and contains ideal gas at pressure $$P_1$$ and temperature $$T_1$$. The other chamber has volume $$V_2$$ and contains ideal gas at pressure $$P_2$$ and temperature $$T_2$$. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

A thin spherical shell of radius $$R$$ has charge $$Q$$ spread uniformly over its surface. Which of the following graphs most closely represents the electric field $$E(r)$$ produced by the shell in the range $$0\le r<\infty ,$$ where $$r$$ is the distance from the center of the shell?

A body is at rest at $$x=0.$$ At $$t=0,$$ it starts moving in the positive $$x$$-direction with a constant acceleration. At the same instant another body passes through $$x=0$$ moving in the positive $$x$$ direction with a constant speed. The position of the first body is given by $$x_1\left(t\right)$$ after time $${}^{\prime }{t}^{\prime };$$ and that of the second body by $$x_2\left(t\right)$$ after the same time interval. Which of the following graphs correctly describes $$\left(x_1-x_2\right)$$ as a function of time $${}^{\prime }{t}^{\prime }$$ ?

In an experiment, electrons are made to pass through a narrow slit of width $${}^{\prime }{d}^{\prime }$$ comparable to their de Broglie wavelength. They are detected on a screen at a distance $${}^{\prime }{D}^{\prime }$$ from the slit (see figure). Which of the following graphs can be expected to represent the number of electrons $${}^{\prime }{N}^{\prime }$$ detected as a function of the detector position $${}^{\prime }{y}^{\prime }(y=0$$ corresponds to the middle of the slit$$)$$

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure). If a strong diffraction peak is observed when electrons are incident at an angle $${}^{\prime }{i}^{\prime }$$ from the normal to the crystal planes with distance $${}^{\prime }{d}^{\prime }$$ between them (see figure), de Broglie wavelength $${\lambda }_{dB}$$ of electrons can be calculated by the relationship ($$n$$ is an integer)

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure). Electrons accelerated by potential $$V$$ are diffracted from a crystal. If $$d=1\stackrel{\circ }{A}$$ and $$i=30^{\circ },V$$ should be about $$\left(h=6.6\times {10}^{-34}Js,m_e=9.1\times {10}^{-31}kg,e=1.6\times {10}^{-19}C\right)$$