Question

During the testing of a new light bulb, a sensor is placed \(52.5 \mathrm{~cm}\) from the bulb. It records a root-mean-square value of \(9.142 \cdot 10^{-7} \mathrm{~T}\) for the magnetic field of the radiation emitted by the bulb. What is the intensity of that radiation at the sensor's location?

Short answer

Based on the given information, we calculated the intensity of the radiation at the sensor's location as approximately \(1.576 * 10^{-17} W/m^2\).

Step-by-step solution

Convert the distance to meters.

The given distance is \(52.5 cm\), which can be converted to meters by dividing by 100. So, the distance in meters is: $$52.5 cm * \frac{1 m}{100 cm} =0.525 m$$

Calculate the Intensity using the formula.

Now we can use the formula mentioned above to calculate the intensity of the radiation at the sensor's location: Intensity = \((c * μ_0 * B_{rms}^2) / 2\) We have B_{rms}= \(9.142 * 10^{-7} T\), c=\(3 * 10^8 m/s\), and μ_0=\(4 * π * 10^{-7}T m/A\). Substitute these values into the formula and solve for Intensity: Intensity = \(\frac{(3 * 10^8\,\mathrm{m/s} * 4 * \pi *10^{-7}\,\mathrm{Tm/A} * (9.142 * 10^{-7}\,\mathrm{T})^2)}{2}\)

Evaluate the expression for Intensity.

Evaluating the expression for Intensity, we get: Intensity = \(\frac{(3 * 10^8\,\mathrm{m/s} * 4 * \pi *10^{-7}\,\mathrm{Tm/A} * (8.35764 * 10^{-13} \,\mathrm{T^2}))}{2}\) Intensity = \(\frac{(12 * \pi * 10^1 * 8.35764 * 10^{-20})}{2}\) Intensity = \(6 * \pi * 8.35764 * 10^{-19} W/m^2\) Intensity ≈ \(1.576 * 10^{-17} W/m^2\) So, the intensity of the radiation at the sensor's location is approximately \(1.576 * 10^{-17} W/m^2\).

Key concepts

Electromagnetic Radiation

Electromagnetic radiation encompasses a broad range of phenomena, all of which manifest as waves of electric and magnetic fields moving together through space at the speed of light, denoted by c. These waves vary in wavelength and frequency and include forms such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The common household light bulb emits visible light, which is a type of electromagnetic radiation within the specific wavelength range that our eyes can detect.

Understanding electromagnetic radiation is crucial because it includes not just the light we can see but also a host of other energy forms that we rely on daily. For instance, the radiation emitted by a light bulb enables us to see our surroundings, while radio waves transmit signals to our phones and radios.

Root-Mean-Square Value of Magnetic Field

The root-mean-square (rms) value of a magnetic field is a form of mathematical average used in physics to describe the magnitude of a varying magnetic field. If the intensity of a light bulb's electromagnetic radiation were plotted over time, it would look like a wave, with the strength of the magnetic field oscillating between positive and negative values.

In this scenario, using the rms value—typically denoted as B_rms—provides a way to express the effective strength of the field for calculations like determining the radiation intensity. Despite the actual magnetic field continually changing as the waves oscillate, the B_rms gives us a single measurable value to use in practical applications, such as calculating the energy conveyed by an electromagnetic wave.

Intensity of Radiation

The intensity of radiation relates to the power that radiates through a unit area and is expressed in units of watts per square meter (W/m²). It's a measure of how much energy is transmitted in the form of electromagnetic waves through a certain area over a specified period.

In our example involving a light bulb, the intensity of radiation at the location of the sensor indicates how much of the light's energy is reaching and being detected by the sensor. Calculating this intensity requires an understanding of both the nature of electromagnetic waves and the relevant physical constants—like the speed of light (c) and the permeability of free space (μ₀).

Physics Problem-Solving

Physics problem-solving involves a systematic approach to understand and calculate various physical quantities. The steps to solve the problem presented here about the light bulb radiation demonstrate this approach. First, the problem is defined and relevant quantities are identified—like the sensor's distance from the light source and the B_rms of the magnetic field.

Next, we apply appropriate physics formulas and constants to find the desired quantity, which in this case is the intensity of radiation. By methodically converting units, plugging in known values, and performing the arithmetic, we can derive a solution. This process showcases not just mathematical acumen, but also a deep understanding of the underlying physical principles governing how the world works.