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Step-by-Step Solution
Step 1: Identify the key information
The electric field of the electromagnetic wave is given by
E = 50 \sin\bigl(\omega (t - x/c)\bigr)\text{ (in N/C)}.
The energy contained in a volume V is 5.5 \times 10^{-12} \text{ J}.
We are also given:
\varepsilon_0 = 8.8 \times 10^{-12}\,\text{C}^2 \text{N}^{-1}\text{m}^{-2} .
We need to find the volume V in \text{cm}^3 .
Step 2: Write down the formula for energy density
The energy density u (energy per unit volume) of an electric field in free space is
u = \tfrac{1}{2} \varepsilon_0 E_0^2,
where E_0 is the amplitude (maximum value) of the electric field.
Step 3: Relate the total energy to volume
The total energy U in volume V is
U = u \times V = \bigl(\tfrac{1}{2} \varepsilon_0 E_0^2\bigr)\,V.
Step 4: Substitute the given values
We know U = 5.5 \times 10^{-12}\,\text{J} , \varepsilon_0 = 8.8 \times 10^{-12} , and E_0 = 50\,\text{N/C} . So,
\[
5.5 \times 10^{-12} = \tfrac{1}{2} \times 8.8 \times 10^{-12} \times (50)^2 \times V.
\]
Step 5: Simplify step by step
Compute E_0^2 = 50^2 = 2500.
Then, \tfrac{1}{2} \times 8.8 \times 10^{-12} \times 2500 = \tfrac{1}{2} \times (8.8 \times 2500) \times 10^{-12}.
8.8 \times 2500 = 22000. Half of 22000 is 11000, so the coefficient becomes 1.1 \times 10^4, hence
\tfrac{1}{2} \times 8.8 \times 10^{-12} \times 2500 = 1.1 \times 10^{-8}.
So the equation becomes
\[
5.5 \times 10^{-12} = \bigl(1.1 \times 10^{-8}\bigr)\, V.
\]
Step 6: Solve for V
\[
V = \frac{5.5 \times 10^{-12}}{1.1 \times 10^{-8}}
= 5 \times 10^{-4} \text{ m}^3
= 0.0005 \text{ m}^3.
\]
Step 7: Convert volume to cm³
Recall that 1 \text{ m}^3 = (100 \text{ cm})^3 = 10^6 \text{ cm}^3. Therefore,
\[
V = 0.0005 \text{ m}^3 \times 10^6 \text{ cm}^3 / \text{m}^3
= 500 \text{ cm}^3.
\]
Final Answer
The volume V is 500 \text{ cm}^3.
