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Step-by-Step Solution
Step 1: Write down the formula for intensity of an electromagnetic wave
The intensity $I$ of a uniform electromagnetic wave is related to its maximum electric field $E_0$ by the formula:
$I = \frac{1}{2}\,\varepsilon_0\,c\,E_0^2
where $I$ is the intensity (power per unit area), $\varepsilon_0$ is the permittivity of free space, and $c$ is the speed of light in vacuum.
Step 2: Calculate the intensity from the given power and area
The laser power $P$ is 27 mW (i.e., $27 \times 10^{-3}$ W), and the cross-sectional area $A$ is $10\,\mathrm{mm}^2$. First, convert the area to square meters:
$10\,\mathrm{mm}^2 = 10 \times 10^{-6}\,\mathrm{m}^2 = 1 \times 10^{-5}\,\mathrm{m}^2.
Hence, the intensity $I$ is:
$I = \frac{P}{A}
= \frac{27 \times 10^{-3}\,\mathrm{W}}{1 \times 10^{-5}\,\mathrm{m}^2}
= 2.7 \times 10^3\,\mathrm{W\,m^{-2}}.
Step 3: Rearrange the intensity relation to solve for $E_0$
From
$
I = \frac{1}{2}\,\varepsilon_0\,c\,E_0^2,
$
we can rearrange to find $E_0$:
$E_0 = \sqrt{\frac{2I}{\varepsilon_0\,c}}.
Step 4: Substitute numerical values and evaluate
Use the given values: $\varepsilon_0 = 9 \times 10^{-12}\,\mathrm{SI\ units}$ and $c = 3 \times 10^8\,\mathrm{m\,s^{-1}}$, and $I = 2.7 \times 10^3\,\mathrm{W\,m^{-2}}$:
$
E_0
= \sqrt{\frac{2 \times (2.7 \times 10^3)}{(9 \times 10^{-12}) \times (3 \times 10^8)}}
.
First compute the denominator:
$(9 \times 10^{-12}) \times (3 \times 10^8) = 27 \times 10^{-4} = 2.7 \times 10^{-3}.
Then the expression inside the square root becomes:
$\frac{2 \times 2.7 \times 10^3}{2.7 \times 10^{-3}} = \frac{5.4 \times 10^3}{2.7 \times 10^{-3}}
= \left(\frac{5.4}{2.7}\right) \times 10^{3 + 3}
= 2 \times 10^6.
Thus,
$
E_0 = \sqrt{2 \times 10^6}\,\mathrm{V\,m^{-1}} = \sqrt{2} \times 10^3\,\mathrm{V\,m^{-1}} \approx 1.4 \times 10^3\,\mathrm{V\,m^{-1}}.
$
This is $1.4\,\mathrm{kV\,m^{-1}}$.
Step 5: State the final answer
The magnitude of the maximum electric field in the electromagnetic wave is approximately $1.4\,\mathrm{kV\,m^{-1}}$.
