Model a torch battery of length \(l\) to be made up of a thin cylindrical bar of radius \('a'\) and a concentric thin cylindrical shell of radius ' \(b\) ' fille in between with an electrolyte of resistivity \(\rho\) (see figure). If the battery is connected to a resistance of value \(R ,\) the maximum Joule heating in \(R\) will take place for

In a double slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance from the plane of slits. If the screen is moved by \(5 \times 10^{-2} \,m\) towards the slits, the change in fringe width is \(3 \times 10^{-3} \,cm\). If the distance between the slits is \(1 \,mm\), then the wavelength of the light will be_______ \(nm\)

The reverse breakdown voltage of a Zener diode is \(5.6\, V\) in the given circuit. The current \(I_z\) through the Zener is......\(mA\)

A tightly wound long solenoid carries a current of 1.5 A . An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns . The number of turns per metre in the solenoid is ________.
[Take mass of electron \(\mathrm{m}_{\mathrm{e}}=9 \times 10^{-31} \mathrm{~kg}\), charge of electron \(\left|\mathrm{q}_{\mathrm{e}}\right|=1.6 \times 10^{-19} \mathrm{C}\),
\(\left.\mu_0=4 \pi \times 10^{-7} \frac{\mathrm{~N}}{\mathrm{~A}^2}, 1 \mathrm{~ns}=10^{-9} \mathrm{~s}\right]\)

A tuning fork of frequency \(480\, Hz\) is used in an experiment for measuring speed of sound \((\nu )\) in air by resonance tube method. Resonance is observed to occur at two successive lengths of the air column, \({\ell _1} = 30\,cm\) and \({\ell _2} = 70\,cm\). Then \(\nu\) is equal to ..... \(ms^{-1}\)

A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The rational between temperature and volume for the process is \(TV^x =\) constant, then \(x\) is

A moving coil galvanometer allows a full scale current of \(10^{-4}\, A\). A series resistance of \(2\, M\,\Omega \) is required to convert the above galvanometer into a voltmeter of range \(0 -5\, V\). Therefore the value of shunt resistance required to convert the above galvanometer into a ammeter of range \(0-10\, mA\) is....\(\Omega \)

If \(\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^{3}}\) is equal to \(L,\) then the value of \((6 L +1)\) is

A poly-atomic molecule ( \(C_V=3 R, C_P=4 R\), where \(R\) is gas constant) goes from phase space point \(\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^5 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5 \times 10^4 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^4\right.\) \(\left.\mathrm{Pa}, \mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^3\right)\). A to B is an adiabatic path and \(B\) to \(C\) is an isothermal path. The net heat absorbed per unit mole by the system is :

In meter bridge experiment for measuring unknown resistance ' \(S\) ', the null point is obtained at a distance \(30 cm\) from the left side as shown at point \(D\). If \(R\) is \(5.6 k \Omega\), then the value of unknown resistance ' \(S\) ' will be \(\Omega .\)

In the given figure an ammeter \(A\) consists of a \(240 \Omega\) coil connected in parallel to a \(10 \Omega\) shunt. The reading of the ammeter is _______ \(\mathrm{mA}\).

Let \(f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}\) be differentiable in \((-\infty, 0) \cup(0, \infty)\) and \(f(1)=1\). Then the value of \(ea\), such that \(f(a)=0\), is equal to ...........

The value of \(\cos \,\frac{\pi }{{{2^2}}}.\cos \,\frac{\pi }{{{2^3}}}{._{..................}}.\cos \,\frac{\pi }{{{2^{10}}}}.\,\sin \,\frac{\pi }{{{2^{10}}}}\) is