Q6DQ - University Physics with Modern Physics

Question

The two sources ${\mathbf{S}}_{\mathbf{1}}\mathbf{ }\mathbf{and}\mathbf{ }{\mathbf{S}}_{\mathbf{2}}$ shown in Fig. 35.3 emit waves of the same wavelength $\mathbf{\lambda }$ and are in phase with each other. Suppose ${\mathbf{S}}_{\mathbf{1}}$ is a weaker source, so that the waves emitted by ${\mathbf{S}}_{\mathbf{1}}$ have half the amplitude of the waves emitted by ${\mathbf{S}}_{\mathbf{2}}$. How would this affect the positions of the antipodal lines and nodal lines? Would there be total reinforcement at points on the antipodal curves? Would there be total cancellation at points on the nodal curves? Explain your answers. 

Step-by-Step Solution

Answer

The positions of the anti-nodal and nodal lines are the same as before. There is reduced reinforcement at points on the antinodal curves and there is no total cancellation at points on the nodal curves.

Step 1: Define the principle of superposition.

The principle of superposition states: 

When two or more waves overlap, the resultant intensity at any point and at any instant is found by adding the intensity that would be produced at the point by the individual waves if each were present alone.

Step 2: Explain the positions of nodal and antipodal, total reinforcement and total cancellation.

Since, the position of antipodal and nodal lines depends upon the wavelength, not on the amplitude. Thus, the positions of antipodal and nodal lines will remain the same.

The intensity $l$ is proportional to the square of the amplitudes i.e., $I\infty A^2$. When light from the two given sources interferes, there will be redistribution of the intensity of light, but at the points of anti-nodal curves, the resultant intensity will be less than what it would have been if the sources had equal amplitudes. Similarly, at points of nodal curves, the intensity will be small but not zero.

Hence, there is no change in the positions of the antipodal lines and nodal lines. There is a reduction in the reinforcement at points on the anti-nodal curves. Also, the total cancellation will not take place at points on the nodal curves.