An object is placed in a medium of refractive index 3. An electromagnetic wave of intensity \(6 \times 10^8 \, \text{W/m}^2\) falls normally on the object and it is absorbed completely. The radiation pressure on the object would be (speed of light in free space = \(3 \times 10^8\) m/s) :
- \(36 \, \text{Nm}^{-2}\)
- \(18 \, \text{Nm}^{-2}\)
- \(6 \, \text{Nm}^{-2}\)
- \(2 \, \text{Nm}^{-2}\)
The Correct Option is C
Approach Solution - 1
To find the radiation pressure on the object due to the electromagnetic wave, we start by understanding the concept of radiation pressure in physics. Radiation pressure is exerted upon any surface due to the exchange of momentum between the object and the incident electromagnetic wave. The formula to calculate radiation pressure \(P\) when the wave is completely absorbed is given by:
\(P = \frac{I}{c}\)
where:
- \(I\) is the intensity of the electromagnetic wave (in this case, \(6 \times 10^8 \, \text{W/m}^2\)).
- \(c\) is the speed of light in the medium. In a vacuum or free space, this is approximately \(3 \times 10^8 \, \text{m/s}\).
Since the medium's refractive index is given as \(3\), the speed of light in the medium \(c_m\) is:
\(c_m = \frac{c}{n} = \frac{3 \times 10^8 \, \text{m/s}}{3} = 1 \times 10^8 \, \text{m/s}\)
Now, substitute the values into the radiation pressure formula:
\(P = \frac{6 \times 10^8 \, \text{W/m}^2}{1 \times 10^8 \, \text{m/s}} = 6 \, \text{Nm}^{-2}\)
Thus, the radiation pressure exerted on the object is \(6 \, \text{Nm}^{-2}\).
Approach Solution -2
The radiation pressure \( P \) is given by:
\(P = \frac{I \cdot \mu}{c},\)
where:
- \( I \) is the intensity of the radiation,
- \( \mu \) is the absorption coefficient (fraction of radiation absorbed, here \( \mu = 1 \) for total absorption),
- \( c \) is the speed of light in a vacuum.
Substitute the given values:
- \( I = 6 \times 10^8 \, \text{W/m}^2 \),
- \( \mu = 3 \),
- \( c = 3 \times 10^8 \, \text{m/s} \).
Calculate \( P \):
\(P = \frac{I \cdot \mu}{c} = \frac{6 \times 10^8 \cdot 3}{3 \times 10^8}.\)
Simplify:
\(P = 6 \, \text{N/m}^2.\)
The Correct answer is: \(6 \, \text{Nm}^{-2}\)
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