Correct Answer - Option 4 : 8.425 mW
Concept:
1) The unit vector normal to the surface/plane ax + by = c is given by:
\({\hat a_n} = \frac{{a \cdot {{\hat a}_x} + b \cdot {{\hat a}_y}}}{{\sqrt {{a^2} + {b^2}} }}\)
Where, a, b, c are constants.
2). For a magnetic field vector given as:
\(\vec H = {H_0}\cos \left( {\omega t - \beta x} \right){\hat a_2}\)
The direction of propagation âp = âx and the direction of magnetic field vector âH = âz
3) The average power of the EM wave is given by:
\({\vec p_{avg}} = \frac{1}{2}\frac{{E_0^2}}{{{\eta _0}}}{\hat a_p} = \frac{1}{2}{\eta _0}H_0^2{\hat a_p}\)
4) Average power passing through a surface s is given by:
\({p_{avg}} = {\vec p_{avg}} \cdot \left( {A\;{{\hat a}_n}} \right)\)
Where A is the given area ân is the unit vector normal to the surface s.
Calculation:
\(\vec H = 0.1\cos \left( {\omega t - \beta x} \right)\)
\({\vec p_{avg}} = \frac{1}{2}{\eta _0}H_0^2{\hat a_p}\)
\({\vec p_{avg}} = \frac{1}{2} \times 120\;\pi \times {\left( {0.1} \right)^2}{\hat a_n}\)
= 0.6 π âx
Area of the square plate A = 10 cm × 10 cm
A = 0.1 m × 0.1 m = 0.01 m2
Unit vectors normal to the plane x + 2y = 1 is:
\({\hat a_n} = \frac{{1 \cdot {{\hat a}_x} + 2 \cdot {{\hat a}_y}}}{{\sqrt {{1^2} + {2^2}} }} = \frac{{{{\hat a}_x} + 2{{\hat a}_y}}}{{\sqrt 5 }}\)
\(\vec s = A \cdot {\hat a_n} = 0.01 \times \left( {\frac{{{{\hat a}_x} + 2{{\hat a}_y}}}{{\sqrt 5 }}} \right)\)
Average power passing through the square plate:
\({p_{avg}} = {\vec p_{avg}} \cdot \vec s\)
\( = \left( {0.6\pi \;{{\hat a}_x}} \right) \cdot \left( {\frac{{0.01}}{{\sqrt 5 }}} \right)\left( {{{\hat a}_x} + 2{{\hat a}_y}} \right)\)
\( = 0.6\pi \times \frac{{0.01}}{{\sqrt 5 }}\)
pavg = 0.00843 ≈ 8.43 mW
