(i) The intensity of a light pulse travelling along a communication channel decreases exponentially with distance x according to the relation I = Ioe–αx, where Io is the intensity at x = 0 and α is the attenuation constant.
Show that the intensity reduces by 75 per cent after a distance of 
(ii) Attenuation of a signal can be expressed in decibel (dB) according to the relation 
. What is the attenuation in dB/km for an optical fiber in which the intensity falls by 50 per cent over a distance of 50 km?
(i)
Given,
We substitute the value of x in the given equation.
Percentage change in intensity is given by:
The negative sign shows reduction. Hence, the intensity reduces by 75%.
(ii)
Let the original intensity be I0 and the new intensity be I.
Now, as the intensity reduces by 50%,
Now,
Using the given formula, the gain in dB is
The negative sign shows that there is attenuation (reduction in intensity).
Now, attenuation (in dB/km) is : 