JEE Advanced Physics Syllabus

Give the working formula for Young's modulus of a wire of radius $r$, length $L$, loaded by mass $m$ producing extension $\ell$.

$$Y=\frac{mgL}{\pi r^{2}\ell}$$ where $A=\pi r^{2}$ is the cross-sectional area and $F=mg$.

What is the SI unit of Young's modulus and its dimensional formula?

SI unit: $\mathrm{N\,m^{-2}}$ (pascal, Pa). Dimensional formula: $[\mathrm{M\,L^{-1}\,T^{-2}}]$ (same as stress/pressure).

Why is the search-method (Searle's) apparatus for Young's modulus designed with two wires?

Two identical wires from the same support eliminate errors due to changes in temperature and yielding/sagging of the support; the reference wire compensates so only the elongation due to load is measured.

State the formula for surface tension determined by the capillary rise method.

$$T=\frac{r h \rho g}{2\cos\theta}$$ For water in glass $\theta\approx 0$ so $\cos\theta\approx1$, giving $T=\dfrac{rh\rho g}{2}$, where $r$ is tube radius and $h$ the rise.

How does adding detergent to water affect its surface tension and capillary rise?

Detergent (a surfactant) lowers the surface tension of water, which decreases the capillary rise $h$ since $h\propto T$.

Why must a capillary tube of fine, uniform bore be used in the capillary-rise experiment?

The rise $h=\dfrac{2T\cos\theta}{r\rho g}$ is inversely proportional to radius $r$; a fine bore gives a measurably large rise, and uniform bore ensures $r$ is constant along the tube.

What is the SI unit and dimensional formula of surface tension?

SI unit: $\mathrm{N\,m^{-1}}$ (or $\mathrm{J\,m^{-2}}$). Dimensional formula: $[\mathrm{M\,T^{-2}}]$.

State the principle of calorimetry (method of mixtures).

When bodies at different temperatures are mixed in an isolated system, heat lost by the hotter body equals heat gained by the colder body: $\text{Heat lost}=\text{Heat gained}$.

Write the working relation for the specific heat $s$ of a liquid using the method of mixtures.

Heat lost by hot solid $=$ heat gained by liquid $+$ calorimeter: $$m_s c_s(\theta_s-\theta)=(m_l c_l+m_c c_c)(\theta-\theta_l).$$ Solve for the unknown $c_l$.

What is the water equivalent of a calorimeter?

The water equivalent $W=m_c c_c$ is the mass of water that would absorb the same amount of heat as the calorimeter for the same rise in temperature; here $c_c$ is the specific heat of the calorimeter material.

State the mirror formula and its sign convention basis.

$$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$$ Using the Cartesian sign convention, distances measured against the incident light are negative; for a concave mirror $f$ is negative.

State the thin lens formula used in the $u$-$v$ method.

$$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$$ For a convex lens, real object and real image give $u$ negative and $v$ positive, yielding a positive focal length $f$.

In the $u$-$v$ method, how is focal length found from a graph of $\frac{1}{v}$ versus $\frac{1}{u}$?

The plot is a straight line whose intercepts on both axes have magnitude $\dfrac{1}{f}$; thus $f$ is the reciprocal of the magnitude of either intercept.

In the $u$-$v$ method, how does a graph of $v$ versus $u$ give the focal length?

The $v$–$u$ curve is a rectangular hyperbola; the line $v=u$ meets it at the point $(2f,2f)$, so the focal length is half that coordinate: $f=\dfrac{\text{coordinate}}{2}$.

State the resonance condition (first and second resonance) for a closed (resonance column) air pipe.

For a closed pipe, resonance lengths are $l_1=\dfrac{\lambda}{4}-e$ and $l_2=\dfrac{3\lambda}{4}-e$, where $e$ is the end correction. Subtracting: $l_2-l_1=\dfrac{\lambda}{2}$.

How is the speed of sound found in the resonance column experiment?

$$v=f\lambda=2f(l_2-l_1)$$ since $\lambda=2(l_2-l_1)$; here $f$ is the tuning-fork frequency and $l_1,l_2$ the first two resonance lengths.

What is the end correction in the resonance column experiment, and how is it obtained?

The end correction $e$ accounts for the antinode forming slightly above the open tube end. $e=\dfrac{l_2-3l_1}{2}$; it is approximately $0.3\,d$, where $d$ is the inner diameter.

State Ohm's law and the condition under which it holds.

At constant temperature, the current through a conductor is directly proportional to the potential difference across it: $V\propto I$, i.e. $V=IR$, where $R$ is constant (the resistance).

In verifying Ohm's law, how are the voltmeter and ammeter connected, and why?

The voltmeter is connected in parallel across the resistor (it has high resistance, drawing negligible current), and the ammeter in series with it (low resistance, minimal voltage drop).

What does the $V$-$I$ graph for an ohmic conductor look like, and how is resistance obtained?

It is a straight line through the origin; the resistance equals the slope: $R=\dfrac{V}{I}=\text{slope of }V\text{ vs }I$.

Define specific resistance (resistivity) and give its relation to resistance.

Resistivity $\rho$ is the resistance of a conductor of unit length and unit cross-sectional area: $R=\dfrac{\rho L}{A}$, so $\rho=\dfrac{RA}{L}$. SI unit: $\Omega\,\text{m}$.

State the balance condition of a metre (Wheatstone) bridge for finding an unknown resistance.

At balance (null deflection), with balancing length $l$ from one end, $$\frac{R}{S}=\frac{l}{100-l}\quad\Rightarrow\quad S=R\,\frac{100-l}{l}.$$

How is the resistivity of a wire computed after finding its resistance $X$ on a metre bridge?

$$\rho=\frac{X\,\pi r^{2}}{L}=\frac{X\pi d^{2}}{4L}$$ where $r$ (or diameter $d$, from screw gauge) is the wire radius and $L$ its length.

What is a post office box and what does it measure?

A post office box is a Wheatstone-bridge-based instrument with ratio arms ($P$, $Q$) and a variable resistance arm ($R$); at balance the unknown $X=\dfrac{Q}{P}R$, used to measure resistance/resistivity.

Distinguish between distance and displacement.

Distance is the total path length travelled (a scalar, always $\geq 0$). Displacement is the shortest straight-line vector from initial to final position; $|\text{displacement}|\leq\text{distance}$.

Define average velocity and instantaneous velocity.

Average velocity $=\dfrac{\Delta \vec{x}}{\Delta t}$. Instantaneous velocity $=\lim\limits_{\Delta t\to 0}\dfrac{\Delta \vec{x}}{\Delta t}=\dfrac{d\vec{x}}{dt}$.

Write the three equations of motion for constant acceleration in one dimension.

$$v=u+at,\qquad s=ut+\tfrac{1}{2}at^{2},\qquad v^{2}=u^{2}+2as.$$

What is the displacement of a uniformly accelerated body in the $n$th second of its motion?

$$s_{n}=u+\frac{a}{2}(2n-1).$$ This gives the distance covered during the $n$th second alone.

For a projectile launched with speed $u$ at angle $\theta$, write the time of flight, maximum height, and horizontal range.

$$T=\frac{2u\sin\theta}{g},\quad H=\frac{u^{2}\sin^{2}\theta}{2g},\quad R=\frac{u^{2}\sin 2\theta}{g}.$$

At what angle is the horizontal range of a projectile maximum, and what is that range?

Range is maximum at $\theta=45^{\circ}$ (since $\sin 2\theta=1$), giving $R_{\max}=\dfrac{u^{2}}{g}$.

What is the shape of a projectile's trajectory, and give its equation.

The path is a parabola: $$y=x\tan\theta-\frac{g x^{2}}{2u^{2}\cos^{2}\theta}.$$

Define angular velocity and relate it to linear speed in circular motion.

Angular velocity $\omega=\dfrac{d\theta}{dt}$ (rad s$^{-1}$). Linear speed $v=r\omega$, where $r$ is the radius of the circular path.

Write the expression for centripetal acceleration in uniform circular motion.

$$a_c=\frac{v^{2}}{r}=\omega^{2}r$$ directed radially inward (toward the centre of the circle).

Distinguish between centripetal and tangential acceleration in non-uniform circular motion.

Centripetal acceleration $a_c=\dfrac{v^{2}}{r}$ points toward the centre and changes the direction of velocity; tangential acceleration $a_t=\dfrac{dv}{dt}$ is along the path and changes the speed. Net $a=\sqrt{a_c^{2}+a_t^{2}}$.