Question

A spherical wave with a wavelength of 2.0 mm is emitted from the origin. At one instant of time, the phase at rrr = 4.0 mm is πradπrad. At that instant, what is the phase at rrr_1 = 3.5 mm ? Express your answer to two significant figures and include the appropriate units.

Answer 1

Complete Question

A spherical wave with a wavelength of 2.0 mm is emitted from the origin. At one instant of time, the phase at r_1 = 4.0 mm is π rad. At that instant, what is the phase at r_2 = 3.5 mm ? Express your answer to two significant figures and include the appropriate units.

Answer:

The phase at the second point is  $\phi _2 = 1.57 \ rad$

Explanation:

From the question we are told that

    The wavelength of the spherical wave is  $\lambda = 2.0 \ mm = \frac{2}{1000} = 0.002 \ m$

    The first radius  is  $r_1 = 4.0 \ mm = \frac{4}{1000} = 0.004 \ m$

     The phase at that instant is  $\phi _1 = \pi \ rad$

     The second radius is  $r_2 = 3.5 \ mm = \frac{3.5}{1000} = 0.0035 \ m$

Generally the phase difference is mathematically represented as

          $\Delta \phi = \phi _2 - \phi _1$

this can also be expressed as

         $\Delta \phi = \frac{2 \pi }{\lambda } (r_2 - r_1 )$

So we have that

   $\phi _2 - \phi _1 = \frac{2 \pi }{\lambda } (r_2 - r_1 )$

substituting values

     $\phi _2 - \pi = \frac{2 \pi }{0.002 } ( 0.0035 - 0.004 )$

    $\phi _2 = \frac{2 \pi }{0.002 } ( 0.0035 - 0.004 ) + 3.142$

   $\phi _2 = 1.57 \ rad$