Question

A flood light is covered with a filter that transmits red light. The electric field of emerging beam is represented by a sinusoidal plane wave \[{E_x} = 36\sin (1.20 \times {10^7}z + 6 \times {10^{15}}t)V/m\]. The average intensity of beam will be
A. \[0.86W/{m^2}\]
B. \[1.72W/{m^2}\]
C. \[3.44W/{m^2}\]
D. \[6.88W/{m^2}\]

Answer 1

Hint:Here to find average intensity we will use the formula \[{I_{avg.}} = \dfrac{{c \times {{\left( {{E_0}} \right)}^2} \times {\varepsilon _0}}}{2}\],where is speed of light, is amplitude of electric field as given in question and is constant value known as epsilon naught.

Complete step-by-step solution:
Variant electric field, \[{E_x} = 36\sin (1.20 \times {10^7}z + 6 \times {10^{15}}t)V/m\]
Amplitude of electric field, \[{E_0} = 36\]
Speed of light, \[c = 3 \times {10^8}\]
Epsilon naught, \[{\varepsilon _0} = 8.85 \times {10^{ - 12}}N{m^2}{C^{ - 2}}\]
So to calculate average intensity, substitute the values as per given in question in the formula \[{I_{avg.}} = \dfrac{{c \times {{\left( {{E_0}} \right)}^2} \times {\varepsilon _0}}}{2}\]
\[\Rightarrow {I_{avg.}} = \dfrac{{3 \times {{10}^8} \times 8.85 \times {{10}^{ - 12}} \times {{\left( {36} \right)}^2}}}{2}\]
\[\Rightarrow {I_{avg.}} = \dfrac{{3 \times 8.85 \times {{10}^{ - 4}} \times 1296}}{2}\]
\[\Rightarrow {I_{avg.}} = \dfrac{{26.55 \times 1296 \times {{10}^{ - 4}}}}{2}\]
\[\Rightarrow {I_{avg.}} = \dfrac{{34408.8 \times {{10}^{ - 4}}}}{2}\]
\[\Rightarrow {I_{avg.}} = 17204.4 \times {10^{ - 4}}\]
\[\Rightarrow {I_{avg.}} = 1.72W/{m^2}\]

Additional Information: Electric charge is conveyed by the subatomic particles of a molecule, for example, electrons and photons. Each charged molecule makes a space around it where the impact of its electric power is felt. This space around the charged particles is known as the Electric field. At whatever point a unit test charge is set in this electric field it will encounter the power produced by the source molecule. The measure of power experienced by a unit charged molecule when it is set in the electric field is known as Electric field force.

Note:- Electromagnetic beam is synchronize oscillations of electric and magnetic fields by moving charges on any axis generally perpendicular to z axis and oscillating alternatively and perpendicularly to each other and the energy carried by any wave is proportional to its amplitude squared where \[{E_0}\] is maximum electric field strength of a sinusoidal wave as given in above question and in above question it is varying with respect to both distance and time.