Units and Measurements - Class 11 Part 1

In a non-linear model of the photoelectric effect, the intense incident light modifies the surface properties of a metal. The maximum kinetic energy $K$ of the emitted photoelectrons is given by the relation: $K = h\nu - \Phi_0 \exp\left(-\frac{I \lambda}{\alpha \nu}\right)$ where $h$ is Planck's constant, $\nu$ is the frequency of the incident light, $\lambda$ is its wavelength in vacuum, $I$ is the intensity of the incident light, $\Phi_0$ is the baseline work function of the material, and $\alpha$ is a positive constant. Assuming the equation is dimensionally consistent, what is the dimensional formula of the constant $\alpha$, and what is the theoretical threshold frequency $\nu_{th}$ in the limit of extremely high light intensity ($I \to \infty$)?

A spherically symmetric volume charge distribution in free space is given by $\rho(r) = \frac{\rho_0}{r} \sin(kr)$ for $0 < r \le R$, and $\rho(r) = 0$ for $r > R$, where $\rho_0$ is a dimensional constant. The constant $k$ is chosen such that the charge density becomes zero for the first time at the surface of the sphere $r = R$. What is the total electric flux through a concentric spherical surface of radius $2R$?

An ideal LC circuit contains a capacitor of constant capacitance $C$ and an inductor whose inductance varies with time as $L(t) = L_0 \ln(1 + \alpha t)$, where $L_0$ and $\alpha$ are positive constants. If $q(t)$ represents the charge on the capacitor at time $t$, which of the following represents the dimensional formula of $\alpha$ and the correct differential equation governing the charge $q(t)$?

A transverse wave packet on a taut string is described by the incident wave equation $Y_i(x,t) = A \exp[-(ax^2 + bt^2 + cxt)]$, where $A, a, b$ are positive constants and $c$ is a negative constant. The wave is incident on a fixed rigid boundary located at $x = 0$. By applying the principle of dimensional homogeneity and boundary conditions, determine the wave propagation speed $v$ and the equation of the reflected wave $Y_r(x,t)$.

In a theoretical model of a quantum fluid, the fundamental units are Planck's constant $h$, the speed of light $c$, and the Planck length $\ell_p$. The fluid's effective viscosity is postulated to be $\eta = \frac{h}{\ell_p^3}$. A microscopic sphere of mass $m$ and radius $r$ falls under gravity $g$ in this fluid (ignoring buoyant forces). Due to a velocity-dependent boundary layer at the Planck scale, the drag force deviates from Stokes' law and is given by $F_d = 6\pi \eta r v \left( 1 + \beta \frac{\ell_p v}{r c} \right)$ where $\beta$ is a dimensionless constant and $v$ is the velocity of the sphere. Assuming the perturbation is small ($\beta \frac{\ell_p v}{r c} \ll 1$), what is the first-order correction to the terminal velocity, $\Delta v_t = v_t - v_{t0}$, where $v_{t0}$ is the unperturbed terminal velocity?

In a non-linear dispersive quantum medium, the energy density $u$ (energy per unit volume) of a transverse wave is given by the relation $u = \eta \hbar^\alpha c^\beta G^\gamma \omega^3 k^3$, where $\omega$ is the angular frequency, $k$ is the wave number, and $\eta$ is a dimensionless constant. The parameters $\hbar, c,$ and $G$ represent the reduced Planck's constant, speed of light in vacuum, and universal gravitational constant, respectively. The dispersion relation of this medium is $\omega^2 = \mu k^5$, where $\mu$ is a positive dimensional constant. Which of the following represents the correct expression for the energy flux (intensity) $I$ of the wave?

A heavy nucleus $X$ undergoes spontaneous fission into two fragments $Y$ and $Z$ according to the reaction $X \rightarrow Y + Z$. The atomic masses are given as $m_X = 236.053 \text{ u}$, $m_Y = 140.91 \text{ u}$, and $m_Z = 94.949 \text{ u}$. Taking the energy equivalent of $1 \text{ u}$ to be $931.5 \text{ MeV}$, calculate the $Q$-value of the reaction. Express your final answer with the correct number of significant figures.

A standard resistor has a resistance $R_0 = 10.00 \, \Omega$ at an initial temperature $T_0 = 20.0 \, ^\circ\text{C}$. It is connected across an ideal DC voltage source of $V = 12.00 \, \text{V}$. The resistor is then placed in a thermal bath at $T_f = 32.4 \, ^\circ\text{C}$. The temperature coefficient of resistance is given as $\alpha = 0.018 \, ^\circ\text{C}^{-1}$. Assuming the voltage remains exactly constant, what is the current in the circuit at the new temperature, calculated with the correct application of significant figure rules?

Satellite $S_1$ orbits a planet at a radial distance $R = 6.4 \times 10^3 \text{ km}$ from its center. Satellite $S_2$ orbits the same planet at a radial distance $R + h$, where $h = 1.4 \text{ km}$. According to Kepler's Third Law, calculate the percentage increase in the orbital period of $S_2$ relative to $S_1$, i.e., $\frac{T_2 - T_1}{T_1} \times 100$. Express your final answer with the correct number of significant figures.

A spherical particle falls through a viscous liquid and reaches terminal velocity. The following measurements are recorded:

Weight of the particle: $W = 7.59 \times 10^{-5} \text{ N}$ Buoyant force on the particle: $F_b = 7.2 \times 10^{-5} \text{ N}$ Viscosity of the liquid: $\eta = 1.0 \times 10^{-1} \text{ Pa}\cdot\text{s}$ Radius of the particle: $r = \frac{5.00}{\pi} \text{ mm}$

Assuming Stokes' law is valid, calculate the terminal velocity of the particle, expressing the result with the correct number of significant figures.

A point charge $Q_1$ is fixed at the origin. The surrounding space is filled with a linear, isotropic dielectric medium whose relative permittivity varies radially as $\epsilon_r(r) = \alpha r$, where $\alpha$ is a known exact constant with units of $\text{m}^{-1}$. A second point charge $Q_2$ is placed at a distance $r = R$ from the origin. The maximum percentage errors in the measurement of $Q_1$, $Q_2$, and $R$ are $1\%$, $1\%$, and $2\%$ respectively. Assuming $\alpha$ and the vacuum permittivity $\epsilon_0$ are exact, what is the maximum percentage error in the calculated magnitude of the electrostatic force on $Q_2$?

A galvanometer of resistance $G$ and temperature coefficient of resistance $\alpha_1$ is converted into an ammeter of range $I$ using a shunt of resistance $S$ and temperature coefficient $\alpha_2$. Assuming the full-scale deflection current $I_g$ of the galvanometer is independent of temperature, which of the following expressions represents the fractional change in the range of the ammeter, $\frac{\Delta I}{I}$, for a small temperature rise $\Delta T$?

A point charge $+Q$ is fixed at the origin of a coordinate system. A small planar sensor of area $A$ is constrained to move along the curve $y = x^3$ (where $x > 0$) in the $xy$-plane. The sensor's surface normal $\hat{n}$ is mechanically maintained parallel to the positive $y$-axis ($\hat{n} = \hat{j}$). The magnitude of the electric field $E$ at the sensor's location is determined experimentally by measuring the electric flux $\Delta \Phi$ through the sensor, its area $A$, and its $x$-coordinate. During an experiment at $x = 1 \text{ m}$, the relative error in measuring the flux is $1.5\%$, and the relative error in measuring the area is $1.0\%$. The absolute error in measuring the $x$-coordinate is $0.02 \text{ m}$. Assuming the errors are independent and add up to give the maximum possible error, what is the maximum percentage relative error in the calculated value of the electric field $E$?

In a Young's Double Slit Experiment, the slit separation is $d = 1.0 \text{ mm}$ and the screen distance is $D = 1.0 \text{ m}$. A traveling microscope is used to measure the position of the central maximum on the screen. The maximum uncertainty in any single position reading is equal to the microscope's least count, $LC = 10 \text{ \mu m}$. Initially, a hollow tube of length $L = 10.0 \text{ cm}$ containing vacuum is placed in front of one slit, and the position of the central maximum is recorded. A gas is then slowly pumped into the tube, and the position is recorded again. If the two recorded readings are exactly identical, what is the maximum possible value of $(\mu - 1)$ for the gas, where $\mu$ is its refractive index?

The radius of a spherical black body is measured as $R = 10.0 \text{ cm}$ using an instrument with a least count of $0.1 \text{ cm}$. Its total radiated thermal power is measured as $P = 100 \text{ W}$ using a wattmeter with a least count of $2 \text{ W}$. Assuming the Stefan-Boltzmann constant $\sigma$ is known exactly and the surroundings are at absolute zero, what is the maximum percentage error in the calculated absolute temperature $T$ of the black body?

A digital ratemeter measures the count rate $R$ of a radioactive sample by counting the number of emissions $N$ over a fixed sampling window $\tau$, such that $R = N/\tau$. The ratemeter's display has a least count of $R_0$ (in counts per second). Due to the statistical nature of radioactive decay, the number of counts $N$ follows a Poisson distribution with standard deviation $\sqrt{N}$. A student determines the decay constant $\lambda$ by observing the expected drop in the count rate, $\Delta R$, over a time interval $\Delta t$ (where $\tau \ll \Delta t \ll 1/\lambda$). For the measurement to be valid, the expected drop $\Delta R$ must be at least equal to the sum of the ratemeter's least count and the Poisson standard deviation of the count rate. If the experiment is performed at an instant when the Poisson standard deviation of the rate exactly equals the least count $R_0$, what is the minimum time interval $\Delta t$ required for a valid measurement?

A velocity spectrometer with a resolution (least count) of $\Delta v = 65 \text{ m/s}$ is used to measure the speed distribution of an ideal gas of molar mass $M = 16 \text{ g/mol}$. The instrument cannot resolve the root-mean-square speed $v_{rms}$ and the average speed $v_{avg}$ as distinct peaks if their difference is less than or equal to $\Delta v$. Taking $\sqrt{3} \approx 1.73$, $\sqrt{\frac{8}{\pi}} \approx 1.60$, and the universal gas constant $R = \frac{25}{3} \text{ J mol}^{-1}\text{K}^{-1}$, what is the maximum temperature $T$ of the gas for which these two speeds remain indistinguishable?

A semiconductor diode is forward-biased at a constant current. The voltage $V$ across the diode is given by $V(T) = \frac{E_g(T)}{e} - \beta T$, where $E_g(T)$ is the temperature-dependent band gap energy, $e$ is the elementary charge, and $\beta = 1.2 \times 10^{-3} \text{ V/K}$. The band gap energy follows the empirical relation $E_g(T) = E_g(0) - \alpha T^2$, where $\alpha = 1.5 \times 10^{-6} \text{ eV/K}^2$. The diode is used as a temperature sensor by measuring $V$ with a digital voltmeter having a least count of $4.2 \text{ mV}$. What is the minimum temperature change $\Delta T$ (the 'blind spot') that the voltmeter can resolve at an operating temperature of $T = 300 \text{ K}$?

An electrometer has a constant instrumental zero-error vector $\vec{E}_{ze}$. To calibrate it, a student places the probe at the center of a solid, uniformly charged non-conducting sphere where the true electric field is known to be zero, and observes a reading of $\frac{k R^2}{20 \epsilon_0} \hat{k}$.

Next, the student uses the electrometer to measure the electric field at the center $C$ of a spherical cavity of radius $R/2$ carved inside a different non-conducting sphere of radius $R$. This sphere has a non-uniform volume charge density $\rho = kr$ (where $r$ is the distance from the sphere's center $O$). The cavity is centered at $C$, which is located at a distance $R/2$ from $O$ along the $+z$ axis.

Assuming the electrometer's orientation is unchanged, what is the reading of the electrometer at $C$?

A large open rectangular tank of length $L = 4 \text{ m}$ contains water of density $\rho = 10^3 \text{ kg/m}^3$. The tank is accelerated horizontally along its length with a constant acceleration $a = 7.5 \text{ m/s}^2$. A pressure gauge is attached at the bottom-front corner of the tank (point A, towards the direction of acceleration). The gauge is calibrated to measure gauge pressure but has a positive zero error of $4 \text{ kPa}$. During the motion, the gauge reads $24 \text{ kPa}$. If a small orifice is suddenly opened at the bottom-rear corner of the tank (point B), what is the velocity of efflux of water relative to the tank? (Assume ideal fluid behavior, $g = 10 \text{ m/s}^2$, and the water does not spill over).

A solid of mass $M$ is heated to a temperature measured as $\theta_H$ by a thermometer and dropped into a calorimeter of combined heat capacity $C$ (including water), initially at a measured temperature of $0^\circ\text{C}$. The final steady temperature of the mixture, measured by the same thermometer, is $\theta$. The thermometer's scale is calibrated correctly at $0^\circ\text{C}$ but has a coefficient of linear expansion $\alpha$. During the mixing, the system loses heat to the surroundings (at $0^\circ\text{C}$) given by $Q_{\text{loss}} = \lambda T_f$, where $T_f$ is the true final temperature and $\lambda$ is a constant. Assuming $\alpha$ is very small, which expression gives the specific heat capacity $S$ of the solid, correct to first order in $\alpha$?

A Wheatstone bridge consists of four arms forming a quadrilateral $ABCD$. A battery of EMF $V = 20\ \text{V}$ and negligible internal resistance is connected across $AC$, with $A$ connected to the positive terminal. Arms $AB$, $BC$, and $CD$ are made of standard resistors, each of resistance $R_0 = 100\ \Omega$. Arm $DA$ contains a semiconductor resistor with a temperature-dependent resistance $R_s = R_0 [1 - \alpha (T - T_0)]$, where $\alpha = 1.2 \times 10^{-4}\ \text{K}^{-1}$ and $T_0$ is the ambient temperature.

A galvanometer of resistance $R_g = 50\ \Omega$ is connected across $BD$ and reads positive for current flowing from $D$ to $B$. The semiconductor dissipates heat to the surroundings at $T_0$ with a thermal conductance $K = 10\ \text{mW/K}$. Due to a mechanical defect, the galvanometer has a constant zero error $I_z$. In the steady state, the galvanometer reads $I_{\text{read}} = 550\ \mu\text{A}$.

Assuming the change in resistance is small ($\Delta R \ll R_0$) so that first-order perturbation is valid, what is the zero error $I_z$?

A non-uniform line charge is placed along the positive x-axis from $x = 0$ to $x = a$. The linear charge density is given by $\lambda = c x^2$, where $c$ is a positive constant. A point charge $q_0$ is placed on the y-axis at $y = a$. Which of the following expressions correctly represents the net electric force vector $\vec{F}$ exerted on $q_0$ by the line charge?

A particle of rest mass $M$ moving with initial momentum $p_0 = \frac{4}{3}Mc$ decays into two identical product particles, each of rest mass $m$. The Q-value of this decay process is given as $Q = mc^2$. What is the maximum possible transverse momentum (perpendicular to the initial direction of motion) of one of the product particles in the laboratory frame?

A particle of mass $m = 1 \text{ kg}$ is subjected to two orthogonal restoring forces such that its position coordinates as a function of time $t$ are given by $x(t) = A \sin(\omega t)$ and $y(t) = A \sin\left(\omega t + \frac{\pi}{3}\right)$, where $A = 2 \text{ m}$ and $\omega = \pi \text{ rad/s}$. What is the radius of curvature of the particle's trajectory at the point where its speed is maximum?

An ideal inductor of inductance $L$ carrying an initial steady current $I_0$ is connected across a time-varying resistor at time $t=0$. The resistance varies with time as $R(t) = \alpha t$, where $\alpha$ is a positive constant. There is no other voltage source in the circuit for $t > 0$. Which of the following correctly identifies the dimensional formula of $\alpha$ and the exact expression for the current $i(t)$ in the circuit at any time $t > 0$?

A self-gravitating fluid planet of constant mass $M$ orbits a star of mass $M_*$ in a circular orbit of radius $a$. The planet's internal density profile is given by $\rho(r) = \frac{M}{\pi R^4} r$, where $r$ is the radial distance from the planet's center and $R$ is its dynamic surface radius. Due to tidal interactions, the planet's radius adjusts to its orbital velocity $v$ such that $R(v) = \frac{k}{v^2}$, where $k$ is a positive constant. The gravitational self-energy of the planet is defined as $U = - \int_0^R \frac{G m(r)}{r} d m(r)$, where $m(r)$ is the mass enclosed within radius $r$. Which of the following represents the rate of change of the planet's self-energy with respect to the orbital radius $a$, $\frac{dU}{da}$?

A solid truncated cone of length $L$ (in $\text{m}$) is made of an isotropic material with Young's modulus $Y$ (in $\text{N/m}^2$) and coefficient of volume expansion $\gamma$ (in $\text{K}^{-1}$). The radii of its circular ends are $R$ and $2R$ (in $\text{m}$). The cone is rigidly fixed between two unyielding walls at an initial temperature $T_0$. If the temperature of the cone is uniformly increased to $T_0 + \Delta T$ (where $\Delta T$ is in $\text{K}$), what is the maximum compressive thermal stress developed in the cone? Assume the material remains in the elastic limit and $\gamma \Delta T \ll 1$.

In a Young's Double Slit Experiment (YDSE), the slit separation is $d = 1.5 \text{ mm}$ and the screen distance is $D = 2.0 \text{ m}$. A monochromatic light of wavelength $\lambda = 6000 \text{ \AA}$ and intensity $I_0 = 10 \text{ W/m}^2$ is used. A transparent thin film of thickness $t = 15.0 \text{ \mu m}$ and initial refractive index $\mu = 1.50$ is placed in front of one of the slits. The position of the central maximum on the screen is measured using a traveling microscope with a least count of $LC = 0.02 \text{ mm}$. If the refractive index of the film is slowly varied while keeping its thickness strictly constant, what is the minimum change in the refractive index ($\Delta \mu$) that can be confidently detected as a shift in the central maximum by this microscope?

A perfectly black spherical body of radius $R \text{ m}$ is maintained at an unknown absolute temperature $T \text{ K}$. The total radiated thermal power is measured to be $P \text{ W}$ using a detector with a least count of $\Delta P \text{ W}$. The radius of the sphere is measured using an instrument with a least count of $\Delta R \text{ m}$. The Stefan-Boltzmann constant $\sigma \text{ W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ is assumed to be exact. Which of the following expressions correctly represents the maximum percentage error in the calculated value of the absolute temperature $T$?

A semiconductor diode is used as a temperature sensor. When driven by a constant forward current, the voltage $V$ across the diode is given by the relation: $V(T) = \frac{E_g(T)}{e} - \eta T$ where $e$ is the elementary charge, $T$ is the absolute temperature, and $\eta = 4.0 \times 10^{-3} \text{ V/K}$. The band gap energy $E_g(T)$ of the semiconductor varies with temperature as: $E_g(T) = E_{g0} - \alpha T^2$ where $E_{g0} = 1.2 \text{ eV}$ and $\alpha = 2.0 \times 10^{-5} \text{ eV/K}^2$. The diode voltage is measured using a digital voltmeter with a least count of $\Delta V = 0.48 \text{ mV}$. What is the minimum temperature change $\Delta T$ (the "blind spot") that this sensor can detect at an ambient temperature of $T = 300 \text{ K}$?

A large open vessel containing an ideal liquid of density $\rho$ accelerates horizontally with a constant acceleration $a = \sqrt{3}g$. A pressure gauge is attached to a small orifice on the wall of the vessel. The gauge was calibrated in a static frame to display the gauge pressure as a liquid head $h_{\text{read}}$ (i.e., it assumes $P_{\text{gauge}} = \rho g h_{\text{read}}$). The gauge has a positive zero error $h_0$ (it reads $h_0$ when exposed to atmospheric pressure). Let $z$ be the perpendicular distance from the orifice to the inclined free surface, $Y$ be the vertical depth of the orifice strictly below the free surface, and $X$ be the horizontal distance from the orifice to the free surface at the same elevation. Let $v$ be the efflux velocity of the liquid relative to the vessel when the orifice is opened. Which of the following statements is INCORRECT?

A solid sample of mass $m$ and unknown specific heat $c$ is heated to an initial measured temperature $T_1'$. It is then dropped into a calorimeter containing a liquid, with the system having a combined heat capacity $S$. The initial measured temperature of the calorimeter and the surrounding environment is $T_2'$. The mixture reaches a maximum measured temperature $T_3'$ after a time $t$. During this mixing phase, heat is lost to the surroundings according to Newton's Law of Cooling, with a cooling constant $k$ (in $\text{J} \cdot \text{K}^{-1} \cdot \text{s}^{-1}$). All temperatures are measured using a thermometer whose scale is printed on a metal strip with a coefficient of linear expansion $\alpha$. The thermometer is perfectly calibrated at $0^\circ\text{C}$, and the true temperature $T$ is strictly proportional to the absolute length of the thermometric liquid column. Assuming $\alpha$ is very small such that terms of order $\alpha^2$ and higher can be neglected, determine the specific heat $c$ of the sample.

A non-uniform rod of length $L$ (in $\text{m}$) is placed on a smooth horizontal table. One end of the rod is attached to a vertical wire with torsional constant $\kappa$ (in $\text{N}\cdot\text{m}\cdot\text{rad}^{-1}$). The linear mass density of the rod varies as $\lambda(x) = \beta x$, where $x$ is the distance from the attached end and $\beta$ is a positive constant (in $\text{kg}\cdot\text{m}^{-2}$). An angular displacement sensor measures the angle of the rod. The sensor has a constant positive zero error $\theta_0$ (in $\text{rad}$), meaning it reads $\theta_0$ when the wire is untwisted and exerts no torque. The rod is rotated until the sensor reads $\phi_0$ (in $\text{rad}$, with $\phi_0 > \theta_0$) and is released from rest. What is the maximum angular speed $\Omega_{max}$ (in $\text{rad}\cdot\text{s}^{-1}$) of the rod during its subsequent motion?

A rigid, non-conducting thin rod of length $L = 4a$ is placed along the positive x-axis with one end at the origin $(0,0,0)$. The rod carries a non-uniform linear charge density $\lambda = c x^2$, where $c$ is a positive constant with units of $\text{C}\cdot\text{m}^{-3}$ and $x$ is the distance from the origin. A point charge $q_0$ of mass $m$ is held stationary at the point $(0, 3a, 0)$. A uniform magnetic field $\vec{B} = B_0 \hat{k}$ exists in the entire region. Determine the net initial force vector $\vec{F}$ experienced by the point charge $q_0$ immediately after it is released. Assume the permittivity of free space is $\epsilon_0$.

A thin, rigid, non-conducting ring of mass $m$ (in $\text{kg}$) and radius $r$ (in $\text{m}$) has a total positive charge $q$ (in $\text{C}$) uniformly distributed over it. The ring is constrained to rotate frictionlessly about its central axis, which is fixed along the $z$-axis. At $t=0$, the ring is rotating with an initial angular velocity $\vec{\omega}_0 = \omega_0 \hat{k}$ (in $\text{rad/s}$). A uniform, time-varying magnetic field $\vec{B}(t) = B_0 \cos(\Omega t) \hat{k} + B_0 \sin(\Omega t) \hat{i}$ (in $\text{T}$) is applied, where $B_0$ and $\Omega$ are positive constants. Ignoring self-inductance, what is the angular velocity vector $\vec{\omega}$ of the ring at time $t = \frac{\pi}{\Omega}$ (in $\text{s}$)?

A water tank has a non-uniform cross-sectional area given by $A(h) = k \sqrt{h}$, where $h$ is the height of the water level from the bottom and $k$ is a positive constant with appropriate SI units. A small hole of area $a$ is opened at the bottom ($h=0$). A student calculates the time $T_0$ required for the water level to drop from $h = H$ to $h = H/2$ by assuming the efflux velocity is exactly given by Torricelli's law, $v = \sqrt{2gh}$. However, the true time $T$ differs because the downward velocity of the free surface is not strictly zero. Assuming $a \ll k\sqrt{H}$, which of the following expressions best represents the fractional error $\frac{T_0 - T}{T_0}$ in the student's calculation?

A moving coil galvanometer has a rectangular coil made of a uniform wire. The material of the wire has a temperature coefficient of resistivity $\alpha_\rho = 4.00 \times 10^{-3} \text{ K}^{-1}$ and a coefficient of linear thermal expansion $\alpha_l = 2.00 \times 10^{-5} \text{ K}^{-1}$. The galvanometer is converted into an ammeter of very large range $I$ (where $I \gg I_g$, $I_g$ being the full-scale deflection current) by connecting a shunt resistor of resistance $S$ in parallel. The shunt resistor has a temperature coefficient of resistance $\alpha_s = 1.00 \times 10^{-4} \text{ K}^{-1}$. Assume the torsional constant of the suspension and the uniform radial magnetic field remain strictly constant. If the temperature of the entire setup increases by $\Delta T = 10.0 \text{ K}$, what is the fractional change in the full-scale range of the ammeter, $\frac{\Delta I}{I}$?

A galvanometer has a coil resistance $G = 99 \, \Omega$ and a full-scale deflection current $I_g$ at $20^\circ\text{C}$. It is converted into an ammeter of range $I$ by connecting a shunt resistor of resistance $S = 1 \, \Omega$ at $20^\circ\text{C}$. The temperature coefficients of resistance for the galvanometer coil and the shunt are $\alpha_G = 4.0 \times 10^{-3} \, ^\circ\text{C}^{-1}$ and $\alpha_S = 1.0 \times 10^{-3} \, ^\circ\text{C}^{-1}$ respectively. Assuming $I_g$ is independent of temperature, what is the relative rate of change of the ammeter's full-scale range with respect to temperature, $\frac{1}{I}\frac{dI}{dT}$, at $20^\circ\text{C}$?

A galvanometer of resistance $G_0$ and a constant full-scale deflection current $I_g$ is converted into an ammeter of range $I_0 = n I_g$ (where $n > 1$) at a reference temperature $T_0$ by connecting a shunt resistor $S_0$ in parallel. The temperature coefficients of resistance for the galvanometer coil and the shunt material are $\alpha_g$ and $\alpha_s$ respectively. If the temperature of the system increases by a small amount $\Delta T$, what is the fractional change in the full-scale range of the ammeter, $\frac{\Delta I}{I_0}$? (Assume $\alpha_g \Delta T \ll 1$ and $\alpha_s \Delta T \ll 1$).

A non-uniform electric field is given by $\vec{E} = E_0 (1 + \alpha z) \hat{k}$, where $E_0$ is an unknown constant and $\alpha = 2.0 \text{ m}^{-1}$ is a known exact constant. A student determines $E_0$ by measuring the electric flux $\Phi$ through an open surface $S$ bounded by the curve $x^2+y^2=R^2$ in the $xy$-plane. The true surface $S$ is a paraboloid defined by $z = R - \frac{x^2+y^2}{R}$ for $z \ge 0$. The student measures the base radius as $R = 1.0 \text{ m}$ with a relative error $e_R = \frac{\Delta R}{R}$, and the flux $\Phi$ with a relative error $e_\Phi$. However, the student incorrectly assumes the surface $S$ is a standard hemisphere of radius $R$ ($z = \sqrt{R^2 - x^2 - y^2}$). Let $e_{true}$ be the true maximum relative error in $E_0$ (derived from the correct surface geometry), and $e_{student}$ be the maximum relative error calculated by the student using their flawed geometric assumption. What is the magnitude of the difference $|e_{student} - e_{true}|$? (Assume the normal to the surface has a positive z-component).

A uniform electric field $\vec{E} = E_0 \hat{k}$ exists in a region. To determine the magnitude $E_0$, a physicist measures the absolute electric flux $\Phi$ through an open, bowl-shaped surface $S$. The surface $S$ is generated by revolving the curve $z = H_0 \left(\frac{r}{R_0}\right)^n$ around the $z$-axis (where $r = \sqrt{x^2+y^2}$, and $H_0, R_0$ are known exact dimensional constants) and is bounded by a flat circular top at a measured height $z = H$. The measured quantities are $\Phi$, $H$, and the dimensionless geometric parameter $n$, with maximum absolute errors $\Delta \Phi$, $\Delta H$, and $\Delta n$ respectively. A student INCORRECTLY assumes that the effective area scales proportionally to the enclosed volume $V = \frac{\pi n R_0^2 H^{(n+2)/n}}{H_0^{2/n} (n+2)}$, leading to a flawed error analysis. All of the following expressions represent INCORRECT derivations for the maximum relative error $\frac{\Delta E_0}{E_0}$ EXCEPT:

An exotic electric field in a region of space is given by $\vec{E} = k r^2 \sin^2\theta \hat{r}$, where $k$ is a constant, $r$ is the radial distance, and $\theta$ is the polar angle in spherical coordinates. A uniform magnetic field $\vec{B} = B_0 \hat{k}$ is also present in the region. A student attempts to determine the constant $k$ by measuring the total outward electric flux $\Phi$ through a closed, slightly deformed spherical surface $S$. The surface $S$ is defined by the equation $r(\theta) = R_0 (1 + \beta \cos^2\theta)$, where $\beta \ll 1$ is a known dimensionless deformation parameter.

To simplify calculations, the student INCORRECTLY assumes that the surface $S$ can be treated as a perfect sphere with an effective uniform radius equal to the solid-angle average radius $R_{avg}$ of the deformed surface. They use this $R_{avg}$ to calculate a value $k_{calc}$ from the measured flux $\Phi$.

Assuming the flux $\Phi$ is measured perfectly, what is the relative error $\frac{k_{calc} - k_{true}}{k_{true}}$ in the student's calculated value to first order in $\beta$?

A stationary nucleus $P$ undergoes spontaneous fission into two nuclei $Q$ and $R$ according to the reaction $P \rightarrow Q + R$. The rest masses of the nuclei are measured as $m_P = 238.050 \text{ u}$, $m_Q = 144.12 \text{ u}$, and $m_R = 93.682 \text{ u}$. The energy equivalent of $1 \text{ u}$ is given as $931.5 \text{ MeV}/c^2$. Four students calculate the $Q$-value of this reaction. Due to various misapplications of the rules of significant figures, they obtain different results. All of the following values represent INCORRECT calculations of the $Q$-value, EXCEPT:

A spherical particle of radius $r = 1.00 \times 10^{-3} \text{ m}$ falls at a constant terminal velocity $v_t$ through a viscous liquid. The measured weight of the particle is $W = 3.56 \times 10^{-3} \text{ N}$ and the measured buoyant force exerted by the liquid is $F_b = 1.4 \times 10^{-3} \text{ N}$. The dynamic viscosity of the liquid is $\eta = 0.796 \text{ Pa}\cdot\text{s}$. Assuming Stokes' law is valid and treating $\pi$ as an exact number, what is the terminal velocity $v_t$ of the particle, expressed with the correct number of significant figures?

A thin, non-conducting spherical shell of radius $R$ is centered at the origin of a coordinate system. The surface charge density at any point on the shell is given by the spatial function $\sigma(r) = \sigma_0 \frac{\sin(kr)}{r}$, where $r$ is the radial distance of the point from the origin, $\sigma_0$ is a constant with appropriate SI units to ensure $\sigma(r)$ has units of $\text{C}\cdot\text{m}^{-2}$, and $k$ is a positive constant. If $\epsilon_0$ is the vacuum permittivity, which of the following expressions correctly represents the total electric flux through a concentric imaginary spherical surface of radius $2R$?

Consider the following Assertion (A) and Reason (R).\ \textbf{Assertion (A):} A stationary particle of rest mass $M$ undergoes spontaneous decay into two particles of rest masses $m_1$ and $m_2$. If the Q-value of the decay, $Q = (M - m_1 - m_2)c^2$, is comparable to the rest mass energies of the products, the kinetic energy $K_1$ of the particle $m_1$ is INCORRECTLY given by the classical relation $K_1 = Q \left( \frac{m_2}{m_1 + m_2} \right)$.\ \textbf{Reason (R):} The Q-value is a scalar representing the total kinetic energy released, but its distribution is constrained by the vector conservation of linear momentum ($\vec{p}_1 + \vec{p}_2 = 0$). In the relativistic regime, the non-linear relation $E^2 = (pc)^2 + (mc^2)^2$ dictates that the exact kinetic energy is $K_1 = \frac{Q(Q + 2m_2 c^2)}{2Mc^2}$.

An ideal capacitor of capacitance $C$ is connected in a closed series loop with a time-varying inductor. The inductance varies with time $t$ as $L(t) = L_0 \ln(1 + \alpha t)$, where $L_0$ is a positive constant and $\alpha$ is a dimensional parameter. Let $q(t)$ represent the instantaneous charge on the capacitor, and assume the current in the loop is given by $I = \frac{dq}{dt}$. Which of the following represents the correct differential equation governing the charge $q(t)$, and what is the dimensional formula for $\alpha$?

A transverse wave packet on a taut string is described by the wave function $y(x,t) = A \exp[-(ax^2 + bt^2 + cxt)]$, where $a$ and $b$ are positive constants, and $c$ is a real constant. The wave propagates in the positive $x$-direction without changing its shape. The string is rigidly fixed at $x = L$. Which of the following represents the reflected wave $y_r(x,t)$?

In a theoretical model of quantum gravity, the vacuum acts as a non-linear dispersive medium. The dispersion relation $\omega(k)$ for a wave satisfies the differential equation: $k \frac{d\omega}{dk} - \omega = \frac{h G}{c^2} k^3$ where $h$ is Planck's constant, $G$ is the gravitational constant, and $c$ is the speed of light. In the linear limit ($k \to 0$), the phase velocity approaches $c$. The energy density of these waves is given by $u = \frac{c^4}{G} k^2$. The energy-density flux is defined as $\Phi_E = u v_g$, where $v_g$ is the group velocity. Let $\Phi_0 = \frac{c^8}{h G^2}$ be the fundamental unit of energy-density flux, and $\tilde{k} = k l_p$ be the dimensionless wave number, where $l_p = \sqrt{\frac{h G}{c^3}}$ is the Planck length. Which of the following represents the total energy-density flux $\Phi_E$ in terms of $\Phi_0$ and $\tilde{k}$?