An air-filled rectangular waveguide of internal dimension $$a\,\,cm\,\, \times \,\,b\,\,cm$$ (a > b) has a cutoff frequency of 6 GHz for the dominant $$T{E_{10}}$$ mode. For the same waveguide, if the cutoff frequency of the $$T{E_{11}}$$ mode is 15 GHz, the cutoff frequency of the $$T{E_{01}}$$ mode in GHz is _____________

Two half-wave dipole antennas placed as shown in the figure are excited with sinusoidally varying currents of frequency 3 MHz and phase shift of $$\pi /2$$ between them (the element at the origin leads in phase). If the maximum radiated E-field at the point P in the x-y plane occurs at an azimuthal angle $${60^ \circ }$$, the distance d (in meters) between the antennas is ___________ .

An n–type silicon sample is uniformly illuminated with light which generates 10²⁰ electron hole pairs per cm³ per second. The minority carrier lifetime in the sample is 1 $$\mathrm\mu$$s.In the steady state, the hole concentration in the sample is approximately 10^x , where x is an integer. The value of x is __________________

A piece of silicon is doped uniformly with phosphorous with a doping concentration of $$10^{16}/cm^2$$. The expected value of mobility versus doping concentration for silicon assuming full dopant ionization is shown below. The charge of an electron is $$1.6\;\times\;10^{-19}\;C$$. The conductivity (in S cm^-1) of the silicon sample at 300 K is _________________.