Question

A radiometer can be used to determine the position of an approaching hot object by measuring the amount of irradiation it detects. Consider a radiometer placed at a distance \(H=0.5 \mathrm{~m}\) from the \(x\)-axis is used to measure the position of an approaching small blackbody object. If the radiometer is measuring \(80 \%\) of the irradiation corresponding to the position of the object directly under the radiometer \((x=0)\), determine the position of the object \(L\).

Short answer

Answer: The position of the hot object along the \(x\)-axis is approximately \(0.25\) meters.

Step-by-step solution

Relate irradiation to the position of the object

We note that the fraction of irradiation detected by the radiometer when the object is at position \(x=L\) is directly proportional to the inverse square of the distance between the object and the radiometer. Let the fraction of irradiation be denoted by \(f\), which is given by: \[f = \frac{1}{(H^2 + L^2)}\]

Use given information to find a relationship with \(L\)

The problem statement says that the radiometer is measuring \(80 \%\) of the irradiation corresponding to the position of the object directly under the radiometer \((x=0)\). The fraction when the object is directly under the radiometer is \(f_{0} = \frac{1}{H^2}\). So we have the equation: \[0.8 f_{0} = \frac{1}{(H^2 + L^2)}\]

Solve the equation for \(L\)

We need to solve the equation from Step 2 for \(L\). Rewriting the equation, we get: \[\frac{1}{(H^2 + L^2)} = \frac{0.8}{H^2}\] Now, we will cross-multiply and then rearrange the equation to isolate \(L^2\): \[(H^2 + L^2) = \frac{H^2}{0.8}\] Subtract \(H^2\) from both sides: \[L^2 = \frac{H^2}{0.8} - H^2\] Finally, take the square root of both sides to get the position \(L\): \[L = \sqrt{\frac{H^2}{0.8}-H^2}\]

Calculate \(L\) using the given value of \(H\)

Now that we have the formula for \(L\), we can plug in the given value for the distance \(H=0.5\) m: \[L = \sqrt{\frac{(0.5)^2}{0.8}-(0.5)^2}\] Calculate the value of \(L\): \[L \approx 0.25 \mathrm{~m}\] Hence, the position of the object \(L\) is approximately \(0.25\) meters from the radiometer.

Key concepts

Radiometry

Radiometry is a field within optics that focuses on measuring electromagnetic radiation, including light. It is concerned with various properties of light, such as its intensity, spatial distribution, and temporal distribution.
This can be particularly useful in various applications, such as determining the position of objects based on the intensity of radiation they emit or reflect.

When using a radiometer, which is an instrument designed to measure the intensity of radiation, you typically analyze data like the amount of irradiation received from a source. Through this, you can infer the source's position or other properties.
In the given exercise, the radiometer is used to measure irradiation from a blackbody object, allowing us to determine its position by analyzing how the detected irradiation changes with distance. The radiometer measures the fraction of the total possible irradiation, enabling us to calculate the distance using established mathematical relations.

Blackbody Radiation

Blackbody radiation refers to the theoretical concept of an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. At thermal equilibrium, a blackbody emits radiation in a characteristic spectrum that depends only on the body's temperature.
The emitted radiation is what we term "blackbody radiation." This principle is fundamental in understanding various physical systems, including stars and simple heating elements.

In the context of the exercise, the small approaching object is considered a blackbody. This means it emits radiation uniformly in all directions with an intensity that is only determined by its temperature. The concept simplifies calculations as it allows us to use known equations without accounting for reflectivity or transmission factors.
For the object's position calculation, the constant emission rate makes it feasible to compare the detected irradiation fraction to determine distances, leveraging the predictable nature of blackbody emission.

Inverse Square Law

The inverse square law is a fundamental principle that describes how a physical quantity goes inversely with the square of the distance from the source of that quantity. In radiometry, it is especially important for understanding how the intensity of light or radiation decreases with distance.
As a blackbody object moves away from or toward the radiometer, the intensity of irradiation detected by the radiometer follows this inverse square law. This means that if you double the distance from the source, the intensity becomes one-fourth.

In the specific exercise, knowing that the radiometer measures 80% of the irradiation as opposed to when the object is directly under it, we applied the inverse square law to relate the distance and the irradiation fraction. This relation was crucial for deriving the position of the blackbody object. By applying the inverse square law, we used the known formula to solve for the distance, given the fraction of detected irradiation.