Question

During the testing of a new light bulb, a sensor is placed \(17.7 \mathrm{~cm}\) from the bulb. It records a root-mean-square value of \(279.9 \mathrm{~V} / \mathrm{m}\) for the electric field of the radiation emitted from the bulb. What is the intensity of that radiation at the sensor's location?

Short answer

Answer: The intensity of the radiation at the sensor's location is approximately \(6.955 \times 10^{-3}\,\frac{\text{W}}{\text{m}^2}\).

Step-by-step solution

Write down the given information

The given information is as follows: - Distance from the light bulb to the sensor: \(17.7\,\text{cm} = 0.177\,\text{m}\) - Root-mean-square value of the electric field: \(279.9\,\frac{\text{V}}{\text{m}}\)

Write the formula for the intensity of electromagnetic waves

The intensity (I) of an electromagnetic wave is related to the electric field (E) and the permittivity of free space (ε₀) by the following formula: \[I = \frac{1}{2} \cdot \varepsilon_0 \cdot c \cdot E^2\] Where ε₀ is the permittivity of free space with a value of \(8.85418782 \times 10^{-12}\,\frac{\text{C}^2}{\text{N}\cdot\text{m}^2}\), c is the speed of light with a value of \(3.0 \times 10^8\,\frac{\text{m}}{\text{s}}\), and E is the electric field in \(\frac{\text{V}}{\text{m}}\).

Substitute the given values and constants into the intensity formula

Now, substitute the given root-mean-square value of the electric field (\(279.9\,\frac{\text{V}}{\text{m}}\)) and constants into the formula for the intensity: \[I = \frac{1}{2} \cdot 8.85418782 \times 10^{-12}\,\frac{\text{C}^2}{\text{N}\cdot\text{m}^2} \cdot 3.0 \times 10^8\,\frac{\text{m}}{\text{s}} \cdot (279.9\,\frac{\text{V}}{\text{m}})^2\]

Calculate the intensity of the radiation at the sensor's location

Perform the calculation to find the intensity of the radiation at the sensor's location: \[I = \frac{1}{2} \cdot 8.85418782 \times 10^{-12}\,\frac{\text{C}^2}{\text{N}\cdot\text{m}^2} \cdot 3.0 \times 10^8\,\frac{\text{m}}{\text{s}} \cdot (279.9\,\frac{\text{V}}{\text{m}})^2 \approx 6.955 \times 10^{-3}\,\frac{\text{W}}{\text{m}^2}\] Therefore, the intensity of the radiation at the sensor's location is approximately \(6.955 \times 10^{-3}\,\frac{\text{W}}{\text{m}^2}\).