Question

A GPR pulse is propagated into a non-magnetic medium comprising a single layer underlain by a half space. If the dielectric constants for the top layer and the half-space are \(\varepsilon_1\) and \(\varepsilon_2\), respectively, the reflection coefficient at normal incidence is

• A. \(\dfrac{\sqrt{\varepsilon_1}-\sqrt{\varepsilon_2}}{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}\)
• B. \(\dfrac{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}{\sqrt{\varepsilon_1}-\sqrt{\varepsilon_2}}\)
• C. \(\dfrac{\sqrt{\varepsilon_1}}{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}\)
• D. \(\dfrac{\sqrt{\varepsilon_2}}{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}\)

Hint:

For GPR at normal incidence in non-magnetic media, replace impedances with \(1/\sqrt{\varepsilon}\); the sign of \(R\) depends on which side has the higher dielectric constant.

Answer 1

The Correct Option is A

Step 1: Intrinsic impedance of a medium.
For electromagnetic waves, the intrinsic impedance is \[ \eta=\sqrt{\frac{\mu}{\varepsilon}}. \] In non-magnetic media \(\mu_1=\mu_2=\mu_0\), so \(\eta\propto 1/\sqrt{\varepsilon}\). Step 2: Normal-incidence reflection coefficient.
At normal incidence, the reflection coefficient at an interface (from medium 1 to 2) is \[ R=\frac{\eta_2-\eta_1}{\eta_2+\eta_1}. \] With \(\eta_i \propto 1/\sqrt{\varepsilon_i}\), \[ R=\frac{\tfrac{1}{\sqrt{\varepsilon_2}}-\tfrac{1}{\sqrt{\varepsilon_1}}}{\tfrac{1}{\sqrt{\varepsilon_2}}+\tfrac{1}{\sqrt{\varepsilon_1}}} =\frac{\sqrt{\varepsilon_1}-\sqrt{\varepsilon_2}}{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}. \] \[ \boxed{R=\dfrac{\sqrt{\varepsilon_1}-\sqrt{\varepsilon_2}}{\sqrt{\varepsilon_1}+\sqrt{\varepsilon_2}}} \]