એક \(EM\) તરંગ હવામાંથી માધ્યમમાં દાખલ થાય છે.તેમના વિદ્યુતક્ષેત્રો અનુક્રમે હવામાં \(\overrightarrow {{E_1}} = {E_{01}}\hat x\;cos\left[ {2\pi v\left( {\frac{z}{c} - t} \right)} \right]\) અને માધ્યમમાં \(\overrightarrow {{E_2}} = {E_{02}}\hat x\;cos\left[ {k\left( {2z - ct} \right)} \right]\) વડે આપવામાં આવે છે.જયાં તરંગ સંખ્યા \(k\) અને આવૃત્તિ \(v\) એ હવાને અનુલક્ષીને છે.માધ્યમ અચુંબકીય છે.જો \(\varepsilon {_{{r_1}}}\) અને \(\varepsilon {_{{r_2}}}\) અનુક્રમે હવા અને માધ્યમની સાપેક્ષ પરમીટીવીટીઓ હોય,તો નીચે આપેલ પૈકી કયો વિકલ્પ સાચો છે?

A \(\frac{{{_{{\epsilon r_1}}}}}{{{_{{\epsilon r_2}}}}} = 2\)
B \(\frac{{{_{{\epsilon r_1}}}}}{{{_{{\epsilon r_2}}}}} = \frac{1}{4}\)
C \(\frac{{{_{{\epsilon r_1}}}}}{{{_{{\epsilon r_2}}}}} = \frac{1}{2}\)
D \(\frac{{{_{{\epsilon r_1}}}}}{{{_{{\epsilon r_2}}}}} = 4\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{{{_{{\epsilon r_1}}}}}{{{_{{\epsilon r_2}}}}} = \frac{1}{4}\)

Step-by-step Solution

Detailed explanation

Velocity of \(EM\) wave is given by \(\mathrm{v}=\frac{1}{\sqrt{\mu \epsilon}}\) Velocity in air \(=\frac{\omega}{\mathrm{k}}=\mathrm{C}\) Velocity in medium \(=\frac{\mathrm{C}}{2}\) Here, \(\mu_{1}=\mu_{2}=1\) as medium is non-magnetic…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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