A cell of internal resistance \(r\) drives current through an external resistance \(R\) . The power delivered by the cell to the external resistance will be maximum when:

A cyclotron is used to accelerate protons. If the operating magnetic field is \(1.0\,T\) and the radius of the cyclotron 'dees' is \(60 cm\), the kinetic energy of the accelerated protons in \(MeV\) will be. [use \(m _{p}=1.6 \times 10^{-27} kg , e =1.6 \times 10^{-19} C\) ]

Steel ruptures when a shear of \(3 .5 \times 10^8\,\,N\,m^{-2}\) is applied. The force needed to punch a \(1\,cm\) diameter hole in a steel sheet \(0.3\,cm\) thick is nearly

A resistor develops \(300 \,J\) of thermal energy in \(15 \,s\), when a current of \(2\, A\) is passed through it. If the current increases to \(3 \,A\), the energy developed in \(10\,\,s\)  is........\(J.\)

If frequency of electromagnetic wave is \(60 \mathrm{MHz}\) and it travels in air along \(\mathrm{z}\) direction then the corresponding electric and magnetic field vectors will be mutually perpendicular to each other and the wavelength of the wave (in \(\mathrm{m}\) ) is _______.

Two concentric circular coils with radii \(1\,cm\) and \(1000\,cm\), and number of turns \(10\) and \(200\) respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be \(.........\times 10^{-8}\,H\)(Take, \(\pi^2=10\))

Given below are two statement  Statement \(-I\) : What \(\mu\) amount of an ideal gas undergoes adiabatic change from state \(\left( P _{1}, V _{1}, T _{1}\right)\) to state \(\left( P _{2}, V _{2}, T _{2}\right)\), the work done is \(W =\frac{1 R \left( T _{2}- T _{1}\right)}{1-\gamma}\), where \(\gamma=\frac{ C _{ P }}{ C _{ V }}\) and \(R =\) universal gas constant, Statement \(-II\) : In the above case. when work is done on the gas. the temperature of the gas would rise. Choose the correct answer from the options given below

A liquid column of height \(0.04 \mathrm{~cm}\) balances excess pressure of soap bubble of certain radius. If density of liquid is \(8 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\) and surface tension of soap solution is \(0.28 \mathrm{Nm}^{-1}\), then diameter of the soap bubble is _______ \(\mathrm{cm}\). \(\text { (if } g=10 \mathrm{~ms}^{-2} \text { ) }\)

The magnetic field at the center of current carrying circular loop is \(B _{1}\). The magnetic field at a distance of \(\sqrt{3}\) times radius of the given circular loop from the center on its axis is \(B_{2}\). The value of \(B_{1} / B_{2}\) will be.

The escape velocity of a body on a planet  \('A'\)  is \(12 \,kms ^{-1}\). The escape velocity of the body on another planet  \('B'\) , whose density is four times and radius is half of the planet  \('A'\), is ................... \(kms ^{-1}\)

Two short bar magnets oflength \(1\ cm\) each have magnetic moments \(1.20\ Am^2\) and \(1.00\ Am^2\) respectively. They are placed on a horizontal table parallel to each other with their \(N\) poles pointing towards the South. They have a common magnetic equator and are separated by a distance of \(20.0\ cm\). The value of the resultand horizontal magnetic induction at the mid-point \(O\) of the line joining their centres is close to  (Horizontal component of earth.s magnetic induction is \(3.6 \times  10^{-5}\) \(Wbm^{-2}\)  )

A source of potential difference \(V\) is connected to the combination of two identical capacitors as shown in the figure. When key ' \(K\) ' is closed, the total energy stored across the combination is \(E _{1}\). Now key ' \(K\) ' is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now \(E _{2}\). The ratio \(E _{1} / E _{2}\) will be :