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Step-by-Step Solution
Step 1: Identify the Key Details
• Power of the bulb, P_{\text{bulb}} = 8\text{ W}
• Efficiency of the bulb, \eta = 10\% = 0.10
• Effective power radiated (since only 10% is radiated), P_{\text{rad}} = \eta \times P_{\text{bulb}} = 0.10 \times 8 = 0.8\text{ W}
• Distance from the bulb, r = 10\text{ m}
• Speed of light in vacuum, c = 3 \times 10^8\text{ m/s}
• Permeability of free space, \mu_0 = 4\pi \times 10^{-7}\text{ H/m}
• We are given the electric field expression at 10 m as E_{\text{peak}} = \frac{x}{10}\sqrt{\frac{\mu_0 c}{\pi}}\text{ V/m} . We need to find x .
Step 2: Express the Intensity of the Wave
For a point source radiating uniformly in all directions, the intensity I at a distance r from the source is:
I = \frac{P_{\text{rad}}}{4\pi r^2} .
Step 3: Relate Intensity to the Electric Field
For an electromagnetic wave in free space, the intensity can also be written as:
I = \frac{1}{2} c \varepsilon_0 E_{\text{peak}}^2 ,
where \varepsilon_0 is the permittivity of free space and E_{\text{peak}} is the peak electric field.
Remember that c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} and \varepsilon_0 = \frac{1}{\mu_0 c^2} .
Step 4: Combine Both Expressions to Find E_{\text{peak}}
Equating the two expressions for intensity:
\frac{P_{\text{rad}}}{4\pi r^2} = \frac{1}{2} c \varepsilon_0 E_{\text{peak}}^2 .
Solving for E_{\text{peak}} , we get:
E_{\text{peak}} = \sqrt{\frac{2\,P_{\text{rad}}}{c \,\varepsilon_0 \,4\pi r^2}} .
Using \varepsilon_0 = \frac{1}{\mu_0 c^2} gives:
E_{\text{peak}} = \sqrt{\frac{2\,P_{\text{rad}}\,\mu_0\,c}{4\pi\,r^2}} .
Step 5: Substitute the Known Values
• P_{\text{rad}} = 0.8\text{ W}
• r = 10\text{ m}
• \mu_0 = 4\pi \times 10^{-7}\text{ H/m}
• c = 3 \times 10^8\text{ m/s}
So,
E_{\text{peak}} = \sqrt{\frac{2 \times 0.8 \,\times\, (4\pi \times 10^{-7}) \,\times\, (3 \times 10^8)}{4\pi \,\times\, 100}} .
Step 6: Compare With the Given Form
The question states that the electric field at 10 m is given by:
E_{\text{peak}} = \frac{x}{10}\sqrt{\frac{\mu_0 c}{\pi}}\text{ V/m} .
By simplifying the above square-root expression numerically and matching it with
\frac{x}{10}\sqrt{\frac{\mu_0 c}{\pi}} , we find:
x = 2 .
Final Answer
\boxed{2} .
