A plane electromagnetic wave falls at right angles to the surface of a plane `-` parallel plate of thickness `l` . The plate is made of non`-` magentic substance whose permittivity decreases exponentially from a value `epsilon_(10` at the front surface down to a value `epsilon_(2)` at the rear one. How lond does it take a given wave phase to travel across this plate gt

Explanation

From the data of the problem the relative permittivity of the medium varies as
`epsilon(x) = epsilon_(1)e^(-(x//l)1n(epsilon_(1))/(epsilon_(2)))`
Hence the local velocity of light
`v(x)=(c)/sqrt(epsilon(x))=(c)/sqrt(epsilon_(1))e^((x)/(2l)1n(epsilon_(1))/(epsilon_(2)))`
Thus the required time `t=int_(0)^(l)(dx)/(v(x))=(sqrt(epsilon_(1)))/( c)int_(0)^(l)e^(-(x)/(2l)1n(epsilon_(1))/(epsilon_(2)))`
`dx=(sqrt(epsilon_(1)))/( c)=(e^(-(1)/(2) 1n(epsilon_(1))/( epsilon_(2)))+1)/( (1)/(2l)1n(epsilon_(1))/(epsilon_(2)))=(2l)/(c)(sqrt(epsilon_(1))-sqrt(epsilon_(2)))/(1n(epsilon_(1))/(epsilon_(2)))`