Embibe Experts Solutions for Chapter: KVPY PROBLEMS 2019 STREAM - SX, Exercise 1: PHYSICS

A short solenoid (length $l$ and radius $r\text{,}$ with $n$ turns per unit length) lies well inside and on the axis of a very long, coaxial solenoid (length $L\text{,}$ radius $R$ and $N$ turns per unit length, with $R>r)$. Current $I$ flows in the short solenoid. Choose the correct statement.

Consider the wall of a dam to be straight with height $H$ and length $L\mathit{.}$ It holds a lake of water of height $h\mathit{(}h\mathit{<}H\mathit{)}$ on one side. Let the density of water be ${\rho }_{\mathrm{w}}$. Denote the torque about the axis along the bottom length of the wall by ${\tau }_1$. Denote also as similar torque due to the water up to height $\frac{h}{2}$ and wall-length $\frac{L}{2}$ by ${\tau }_2.$ Then $\frac{{\tau }_1}{{\tau }_2}$ (ignore atmospheric pressure) is $$

Two containers $C_1$ and $C_2$ of volumes $V$ and $4V$, respectively hold the same ideal gas and are connected by a thin horizontal tube of negligible volume with a valve which is initially closed. The initial pressures of the gas in $C_1$ and $C_2$ are $P$ and $5P$, respectively. Heat baths are employed to maintain the temperatures in the containers at $300 \mathrm{K}$ and $400 \mathrm{K}$, respectively. The valve is now opened. Select the correct statement.

Four electrons, each of mass $m_{\mathit{e}}$ are in a one dimensional box of size $L$. Assume that the electrons are interacting, obey the Pauli exclusion principle and are described by standing de Broglie waves confined within the box. Define $\alpha =\frac{h^2}{8m_e{L}^2}$ and $U_0$ to be the ground state energy. Then

A circuit consists of a coil with inductance $L$ and an uncharged capacitor of capacitance $C$. The coil is in a constant uniform magnetic field such that the flux through the coil is $\varphi$.  At time $t=0$, the magnetic field is abruptly switched off. Let ${\omega }_0=\frac{1}{\sqrt{LC}}$ and ignore the resistance of the circuit. Then, $$ 

Consider the configuration of a stationary water tank of cross-section area $A_0$ and a small bucket as shown in figure. $$

What should be the speed, $v$ of the bucket so that the water leaking out of a hole of cross-sectional area $A$ (as shown) from the water tank does not fall outside the bucket? Take $h=5 \mathrm{m}\text{, }H=5 \mathrm{m}\text{,} g=10 \mathrm{m} {\mathrm{s}}^{-2}$, $A=5 {\mathrm{cm}}^2$ and $A_0=500 {\mathrm{cm}}^2$.

The circuit below is used to heat water kept in a bucket.

Assume heat loss only by Newton's law of cooling, the variation in the temperature of the water in the bucket as a function of time is depicted by,

A bubble of radius $R$ in water of density $\rho$ is expanding uniformly at speed $v$. Given that water is incompressible, the kinetic energy of water being pushed is,