Light Intensity and the Inverse Square Law (HSC SSCE Physics): Revision Notes

Light Intensity and the Inverse Square Law

Understanding light intensity

When light emanates from a point source, it radiates uniformly in all directions into the surrounding space. As the light travels further from its source, the same amount of energy becomes distributed across increasingly larger areas. This fundamental behaviour of light leads to an important relationship between distance and intensity.

Light intensity ($I$) is defined as the energy per second passing through each unit of surface area. It is measured in watts per square metre ($\text{W m}^{-2}$). The intensity depends on two key factors: the strength of the source and the distance from that source.

The inverse square law

Derivation of the law

Consider a point source of light with source strength $S$. The source strength represents the total energy emitted per second by the source, measured in watts ($\text{W}$) or joules per second ($\text{J s}^{-1}$).

The light spreads outward in all directions, forming an expanding sphere. At any distance $r$ from the source, the light is distributed over the surface of a sphere with area:

$A = 4\pi r^2$

The intensity at this distance is the source strength divided by the surface area:

$I = \frac{S}{4\pi r^2}$

This can be rearranged to show:

$I = \frac{1}{4\pi} \times \frac{S}{r^2}$

From this equation, we can identify two important proportional relationships:

The constant $\frac{1}{4\pi}$ in the formula indicates that the light spreads over the surface of a sphere. This inverse square relationship appears in other areas of physics, including gravitational fields, electrostatic forces, and sound intensity.

Comparing intensities at different distances

When working with practical problems, we often need to compare the intensity of light at two different distances from the same source. Using the inverse square law:

$I_1 = \frac{S}{4\pi r_1^2} \quad \text{and} \quad I_2 = \frac{S}{4\pi r_2^2}$

Taking the ratio of these intensities:

$\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} = \left(\frac{r_2}{r_1}\right)^2$

Rearranging this gives the useful formula:

$I_1 r_1^2 = I_2 r_2^2$

Worked examples

Investigation: Testing the inverse square law

Aim

To measure the variation in light intensity with distance from the source and verify the inverse square law experimentally.

Hypothesis

Students should write a hypothesis predicting that light intensity will vary inversely with the square of the distance from the source. For example: "The intensity of light will decrease proportionally to the square of the distance from the source."

Materials

• Point source of light (such as a light globe)
• Black curtain or fabric
• Measuring tape
• Light meter (or suitable smartphone app) to measure light intensity
• Graph paper or computer with spreadsheet program

Risk assessment

What are the risks in doing this investigation?	How can you manage these risks to stay safe?
The room will be darkened, so items on the floor may present trip hazards.	Remove any potential trip hazards.
Staring at the light source may temporarily hurt your eyes.	Avoid staring directly at the light source.

Method

1. Darken the room as much as possible to reduce ambient light interference.
2. Place the point source of light in front of a black curtain or fabric. This prevents reflected light from walls or other surfaces from reaching the light meter, ensuring you measure only direct light from the source.
3. Set up the light meter at a measured distance from the source.
4. Record measurements of light intensity at approximately 10 different distances, ranging from $0.20 \text{ m}$ to $5.0 \text{ m}$.
5. For each measurement, record the distance $d$ and the corresponding intensity $I$.
6. The actual units for intensity readings from your light meter are not critical - relative values are sufficient for this investigation.

Recording results

Record your data in a table with the following column headings:

$d$ (m)	$I$ (units)	$\frac{1}{d}$ (m$^{-1}$)	$\frac{1}{d^2}$ (m$^{-2}$)

After collecting your data, calculate the values for the third and fourth columns:

• $\frac{1}{d}$ is the reciprocal of the distance
• $\frac{1}{d^2}$ is the reciprocal of the distance squared

Analysis of results

Discussion points

When writing up your investigation, consider:

1. Do your results support the inverse square law? If your graph shows a linear relationship between $I$ and $\frac{1}{d^2}$, this confirms the inverse square law. Discuss how well your data points fit a straight line.

2. Comment on the scatter in your data points. Perfect alignment is unlikely due to experimental uncertainties. Identify possible sources of error such as:

• Ambient light not completely eliminated
• Reflected light from surfaces
• Imprecise distance measurements
• Variations in light source output

3. Suggest improvements to reduce errors:

• Use a darker room or better light shields
• Take multiple readings at each distance and calculate averages
• Ensure the light meter is perpendicular to the light rays
• Use a more stable light source
• Increase the range of distances measured

Conclusion

Your conclusion should reference your hypothesis and state whether the experimental data supports the inverse square law. A good conclusion might read: "The investigation supports the inverse square law for light intensity. The graph of intensity versus $\frac{1}{d^2}$ shows a linear relationship, confirming that light intensity is inversely proportional to the square of the distance from the source."

Applications and significance

The inverse square law has important practical applications:

1. Photography and lighting design: Understanding how light intensity decreases with distance helps photographers and lighting designers position lights effectively. Moving a light source twice as far away reduces the illumination to one-quarter of its original intensity.

2. Astronomy: The inverse square law allows astronomers to calculate the luminosity of stars from their observed brightness. It also explains why planets further from the Sun receive less solar radiation, as demonstrated in the Mars example above.

3. Safety considerations: High-beam headlights can dazzle oncoming drivers when close, but become less problematic at greater distances. When a car is several hundred metres away, the light intensity reaching your eyes is dramatically reduced according to the inverse square law. At 10 times the distance, the intensity is reduced to $\frac{1}{100}$ of its close-range value.

4. Other forms of radiation: The inverse square law applies not only to visible light but to other forms of electromagnetic radiation, including radio waves, X-rays, and gamma rays. It also applies to sound intensity and gravitational and electrostatic forces, making it one of the fundamental relationships in physics.