Final answer:
The ratio of intensities for a path difference of 0 is the square of the sum of the amplitudes squared over the sum of the separate amplitudes squared. For a path difference of λ/4, exact intensity ratios require advanced wave theory and are not addressed with simple amplitude ratios.
Step-by-step explanation:
To find the ratio of intensities at two points on a screen in Young's double slit experiment when waves from the two slits have a path difference of 0 and λ/4, consider the formula for intensity I in terms of amplitude A: I = A2.
For a path difference of 0, the waves are in phase and we get constructive interference, meaning the amplitudes add up. So, if we have amplitudes in the ratio 3:4, the resultant amplitude is 3 + 4 = 7. The ratio of intensities of the maximum (bright fringe) to minimum (dark fringe) is the square of the ratio of amplitudes, so (72)/(32 or 42) which equals 49/9 or 49/16.
For a path difference of λ/4, the waves are out of phase and interfere destructively to some degree. However, precise calculation of intensity for this particular path difference requires knowledge of the phase relationships of waves, which goes beyond simple ratios and involves more advanced wave theory.