NEB Class 12 Science Physics (Technical Stream) Question Paper 2082 Nepal

The moment of linear momentum about a point is angular momentum ($\\vec{L} = \\vec{r} \\times \\vec{p}$).

A liquid does not wet a solid surface when the angle of contact is greater than 90^\\circ (obtuse).

A constant-volume process is isochoric (isovolumetric).

By the first law of thermodynamics, $\\Delta U = Q + W_{on}$ (work done on the system). So $Q = \\Delta U - W_{on} = 80 - 50 = 30\\,\\text{J}$ supplied to the system, i.e. $+30\\,\\text{J}$.

The reciprocal of wavelength, $\\dfrac{1}{\\lambda}$, is the wave number.

In a stationary wave consecutive nodes are separated by $\\dfrac{\\lambda}{2}$.

Diamagnetic materials have a small negative magnetic susceptibility, so $\\chi < 0$.

Kirchhoff's voltage law (loop rule) is a statement of conservation of energy around a closed loop.

The photoelectric (saturation) current is directly proportional to the intensity of the incident radiation, so the graph of $I_p$ versus $I$ is a straight line passing through the origin — graph (C).

Since $\\lambda = \\dfrac{h}{\\sqrt{2mE}}$, $\\lambda \\propto \\dfrac{1}{\\sqrt{E}}$. Increasing $E$ four times ($E \\to 4E$) gives $\\lambda \\to \\dfrac{\\lambda}{\\sqrt{4}} = \\dfrac{\\lambda}{2}$, i.e. $\\dfrac{1}{2}$ times.

Under reverse bias the depletion region widens, so the thickness of the depletion layer increases.

Moment of inertia of a body about an axis is the sum of the products of the masses of its particles and the squares of their perpendicular distances from the axis of rotation: $I = \\sum m_i r_i^2$. It is the rotational analogue of mass and measures a body's opposition to change in its state of rotational motion.

Two factors on which moment of inertia depends:

1. The mass of the body and how that mass is distributed.
2. The position and orientation of the axis of rotation (distance of the mass from the axis).

Consider a thin uniform rod of mass $M$ and length $L$ with linear mass density $\\lambda = \\dfrac{M}{L}$. Take the axis through the centre $O$, perpendicular to the rod. A small element of length $dx$ at distance $x$ from $O$ has mass $dm = \\lambda\\,dx = \\dfrac{M}{L}\\,dx$.

Its contribution to the moment of inertia is $dI = x^2\\,dm = \\dfrac{M}{L}x^2\\,dx$.

Integrating from $x = -\\dfrac{L}{2}$ to $x = +\\dfrac{L}{2}$:

Hence $I = \\dfrac{ML^2}{12}$.

Principle of conservation of angular momentum: If the net external torque acting on a system is zero, the total angular momentum of the system remains constant in magnitude and direction. That is, when $\\tau_{ext} = 0$, $\\vec{L} = I\\omega = \\text{constant}$, so $I_1\\omega_1 = I_2\\omega_2$.

Surface tension of a liquid is the force acting per unit length on either side of an imaginary line drawn tangentially on the free surface of the liquid, the force being perpendicular to the line and along the surface: $T = \\dfrac{F}{L}$. Its SI unit is $\\text{N m}^{-1}$.

Consider a liquid film stretched on a rectangular frame with a movable wire of length $l$. Because the film has two surfaces, the surface-tension force pulling the wire is $F = T \\times 2l$.

If the wire is moved out by a small distance $x$, the work done against surface tension is

where $\\Delta A = 2l\\,x$ is the increase in total surface area (both faces).

This work is stored as surface energy $E$. Therefore the surface energy per unit area is

Hence the surface tension is numerically equal to the surface energy per unit area (their dimensions $\\text{N m}^{-1}$ and $\\text{J m}^{-2}$ are equivalent).

Capillarity is the phenomenon of rise or fall of a liquid in a narrow tube (capillary) when it is dipped in the liquid, caused by surface tension. Liquids that wet the tube (e.g. water in glass) rise, while liquids that do not wet (e.g. mercury in glass) are depressed.

Two suitable examples:

1. Rise of oil in the wick of a lamp/stove.
2. Absorption of water by blotting paper or by the roots of plants through the soil.

Newton assumed that the propagation of sound in a gas is an isothermal process, giving $v = \\sqrt{\\dfrac{P}{\\rho}}$. This underestimated the speed of sound in air by about 16%.

Laplace's correction: Laplace pointed out that sound travels so fast that the compressions and rarefactions occur adiabatically (no heat exchange with surroundings), not isothermally.

For an adiabatic process $PV^{\\gamma} = \\text{constant}$. Differentiating, $dP\\,V^{\\gamma} + P\\gamma V^{\\gamma-1}\\,dV = 0$, which gives the adiabatic bulk modulus $K = -V\\dfrac{dP}{dV} = \\gamma P$.

Replacing the isothermal bulk modulus $P$ by the adiabatic bulk modulus $\\gamma P$, the corrected speed of sound is

where $\\gamma = \\dfrac{C_p}{C_v}$. This gives values in close agreement with experiment.

Harmonics are the frequencies of vibration of a sounding body that are integral multiples of the fundamental (lowest) frequency. If the fundamental frequency is $f_0$, the harmonics are $f_0, 2f_0, 3f_0, \\dots$, where $f_0$ is the first harmonic, $2f_0$ the second harmonic, and so on.

In an open organ pipe both ends are open, so each end is an antinode of displacement. Let $L$ be the length of the pipe and $v$ the speed of sound.

First mode (fundamental, first harmonic): There is one node in the middle and antinodes at both ends, so $L = \\dfrac{\\lambda_1}{2}$, giving $\\lambda_1 = 2L$ and

Second mode (first overtone, second harmonic): $L = \\lambda_2$, so $\\lambda_2 = L$ and

Third mode (second overtone, third harmonic): $L = \\dfrac{3\\lambda_3}{2}$, so $\\lambda_3 = \\dfrac{2L}{3}$ and

Thus the frequencies are in the ratio $f_1 : f_2 : f_3 = 1 : 2 : 3$, i.e. an open organ pipe produces all the harmonics (both even and odd), giving a richer quality of sound.

End correction is the small length that must be added to the measured (geometrical) length of an organ pipe to account for the fact that the displacement antinode at an open end lies slightly outside the open end rather than exactly at it. For a pipe of radius $r$, the end correction is approximately $e = 0.6\\,r$, so the effective length becomes $L_{eff} = L + e$ (for each open end).

Diffraction of light is the bending of light around the edges of an obstacle or aperture and its spreading into the region of geometrical shadow, when the size of the obstacle or aperture is comparable to the wavelength of the light.

The necessary condition for appreciable diffraction is that the size of the obstacle or aperture must be of the same order as (comparable to) the wavelength of the light. The smaller the aperture/obstacle (closer to $\\lambda$), the more pronounced the diffraction.

i) Coherent sources: Two sources are coherent if they emit waves of the same frequency (and wavelength) and have a constant (zero or fixed) phase difference between them with time. Such sources are needed to obtain a steady, observable interference pattern.

ii) Conditions for interference (path difference $\\Delta$):

Constructive interference (bright fringe): the path difference is a whole number of wavelengths,

equivalently a phase difference of $2n\\pi$.

Destructive interference (dark fringe): the path difference is an odd multiple of half wavelengths,

equivalently a phase difference of $(2n-1)\\pi$.

(First alternative)

a) Principle of potentiometer: When a steady current flows through a uniform wire of constant cross-section, the potential difference across any portion of the wire is directly proportional to its length: $V \\propto l$, i.e. $V = kl$ where $k$ (potential gradient) is constant.

b) A potentiometer is preferred over a voltmeter because at the balance point no current is drawn from the cell, so it measures the true emf (the cell's internal resistance causes no potential drop). A voltmeter always draws some current, so it reads the terminal potential difference $V = E - Ir$, which is less than the emf $E$. Hence the potentiometer acts as an ideal voltmeter of infinite resistance.

c) Kirchhoff's laws:

1. Junction (current) law: The algebraic sum of currents meeting at a junction is zero, $\\sum I = 0$ (conservation of charge).
2. Loop (voltage) law: In any closed loop, the algebraic sum of the products of current and resistance plus the algebraic sum of emfs is zero, $\\sum IR = \\sum E$ (conservation of energy).

(Second alternative)

a) Biot–Savart's law: The magnetic field $dB$ produced at a point by a small current element $I\\,d\\vec{l}$ is directly proportional to the current $I$, the length $dl$, the sine of the angle $\\theta$ between the element and the line joining it to the point, and inversely proportional to the square of the distance $r$:

b) For a circular coil of radius $r$ carrying current $I$, at the centre every element is at distance $r$ with \\theta = 90^\\circ. So $dB = \\dfrac{\\mu_0}{4\\pi}\\dfrac{I\\,dl}{r^2}$. Integrating around the loop ($\\int dl = 2\\pi r$):

For $N$ turns, $B = \\dfrac{\\mu_0 N I}{2r}$.

A galvanometer is converted into an ammeter by connecting a low resistance (shunt) in parallel with it, so that most of the current to be measured passes through the shunt and only a small fixed fraction passes through the galvanometer.

Let $G$ be the galvanometer resistance, $I_g$ the full-scale deflection current, and $I$ the maximum current to be measured. The shunt $S$ carries $(I - I_g)$. Since the galvanometer and shunt are in parallel, the potential differences are equal:

With this shunt the instrument reads currents up to $I$, and its effective resistance becomes very small.

An ammeter measures the current flowing through a circuit element, and the same current must pass through the ammeter; this is possible only if it is connected in series with that element.

An ideal ammeter has very low (ideally zero) resistance, so connecting it in series does not appreciably change the current it is meant to measure. If it were connected in parallel, its low resistance would short-circuit the element, draw a very large current, and could damage the meter — and it would not measure the branch current. Hence an ammeter is always connected in series.

(First alternative)

a) Einstein's photoelectric equation: When a photon of energy $h\\nu$ strikes a metal surface, part of it ($W_0 = h\\nu_0$, the work function) is used to free the electron and the rest appears as the maximum kinetic energy of the emitted electron:

Threshold frequency $\\nu_0$ is the minimum frequency of incident radiation below which no photoelectrons are emitted, however intense the radiation; $W_0 = h\\nu_0$.

b) Maximum K.E. = photon energy − work function:

(Second alternative)

a) Rectifier: A rectifier is a device (using semiconductor diodes) that converts alternating current (AC) into direct (unidirectional) current (DC).

b) Full-wave rectifier (two diodes, centre-tapped transformer): The secondary of a centre-tapped transformer feeds two diodes $D_1$ and $D_2$ whose outputs are joined to a common load $R_L$ connected to the centre tap. During the positive half-cycle, the upper half makes $D_1$ forward-biased ($D_2$ reverse-biased) and current flows through $R_L$. During the negative half-cycle, the lower half makes $D_2$ forward-biased ($D_1$ reverse-biased) and current again flows through $R_L$ in the same direction. Thus both halves of the AC input are utilised, giving a unidirectional (pulsating DC) output.

c) Waveforms: The input is a full sinusoidal AC wave; the output is a series of positive half-sine pulses for every half cycle (both positive and negative input halves appear as positive output humps).

Simple harmonic motion (SHM) is the to-and-fro periodic motion of a body about a fixed (mean) position in which the restoring force (or acceleration) is directly proportional to the displacement from the mean position and is always directed towards it: $a = -\\omega^2 y$.

Given $Y = a\\sin(\\omega t - kx)$. For a fixed point ($x$ constant), differentiate with respect to time.

Velocity: $\\dfrac{dY}{dt} = a\\omega\\cos(\\omega t - kx)$.

Acceleration: $\\dfrac{d^2Y}{dt^2} = -a\\omega^2\\sin(\\omega t - kx) = -\\omega^2 Y.$

Hence the acceleration is

showing it is proportional to $-Y$ (characteristic of SHM).

Let a mass $m$ hang from a helical spring of force constant $k$. At equilibrium the spring is stretched by $e$, so $mg = ke$.

Now pull the mass down a further small displacement $y$ and release. The spring extension is $(e + y)$, so the upward restoring force is $k(e + y)$, and the net force on the mass is

Thus the restoring force $F = -ky$ is proportional to displacement $y$ and directed towards the mean position — the condition for SHM. Acceleration $a = \\dfrac{F}{m} = -\\dfrac{k}{m}y = -\\omega^2 y$, with $\\omega = \\sqrt{\\dfrac{k}{m}}$.

Hence the motion is simple harmonic, with time period

The root-mean-square (rms) value of an alternating current is that steady (direct) current which produces the same heating effect in a given resistance, over a complete cycle, as the alternating current does. It equals the square root of the mean of the squares of the instantaneous current values over one cycle.

For a sinusoidal alternating current, the rms value is related to the peak (maximum) value $I_0$ by

Similarly $V_{rms} = \\dfrac{V_0}{\\sqrt{2}}$.

Let an alternating voltage $V = V_0\\sin\\omega t$ be applied across a pure inductor of inductance $L$.

The back emf is $-L\\dfrac{dI}{dt}$, and since there is no resistance, $V = L\\dfrac{dI}{dt}$:

Integrating:

Thus $I = I_0\\sin\\!\\left(\\omega t - \\dfrac{\\pi}{2}\\right)$ with $I_0 = \\dfrac{V_0}{\\omega L}$ and inductive reactance $X_L = \\omega L$. The current lags the voltage by a phase angle $\\dfrac{\\pi}{2}$ (90°).

Phasor diagram: the voltage phasor $V_0$ leads the current phasor $I_0$ by 90^\\circ — $V$ is drawn 90^\\circ ahead of (anticlockwise from) $I$.

(First alternative)

a) Cross field: A cross field (crossed fields) is an arrangement in which a uniform electric field and a uniform magnetic field are applied mutually perpendicular to each other (and to the velocity of a charged particle), as used in a velocity selector. The particle goes undeflected when $qE = qvB$, i.e. $v = \\dfrac{E}{B}$.

b) Circular motion of an electron in a magnetic field: When an electron of charge $e$ enters a uniform magnetic field $B$ with velocity $v$ perpendicular to $B$, the magnetic force $F = evB$ is always perpendicular to $v$. A force constant in magnitude and always perpendicular to the velocity causes uniform circular motion. The force provides the centripetal force:

Time period: $T = \\dfrac{2\\pi r}{v} = \\dfrac{2\\pi m}{eB}$ — independent of speed and radius.

c) For the same speed $v$ and field $B$, $r = \\dfrac{mv}{qB}$, so $r \\propto m$ (since $|q|$ is equal for electron and proton). Hence

The proton's radius is about 1836 times that of the electron.

d) For the drop to stay suspended, electric force = weight:

With $q = e = 1.6\\times10^{-19}\\,\\text{C}$, $E = 3\\times10^4\\,\\text{V/m}$, $\\rho = 1000\\,\\text{kg/m}^3$, $g = 9.8\\,\\text{m/s}^2$:

(Second alternative)

a) Ionization potential: The ionization potential of an atom is the minimum accelerating potential (in volts) that must be applied to an electron so that, on collision, it can just remove the most loosely bound electron completely from the atom (i.e. excite it from the ground state to $n = \\infty$).

b) The Balmer series of hydrogen lies in the visible region. Its longest wavelength corresponds to the transition $n = 3 \\to n = 2$:

(Note: the printed value $R = 1.097\\times10^{-7}$ appears to be a misprint for $R = 1.097\\times10^{7}\\,\\text{m}^{-1}$.)

c) Quality of X-rays: The quality (penetrating power / hardness) of X-rays is controlled by the accelerating (tube) potential applied across the X-ray tube. Increasing the tube voltage increases the kinetic energy of the electrons striking the target, producing more energetic (shorter-wavelength) hard X-rays of greater penetrating power; lowering it gives soft X-rays. (The intensity, separately, is controlled by the filament current.)

d) de-Broglie wavelength $\\lambda = \\dfrac{h}{\\sqrt{2mE}}$, with $E = 200\\,\\text{eV} = 200 \\times 1.6\\times10^{-19} = 3.2\\times10^{-17}\\,\\text{J}$: