Exercises: Irradiance

Irradiance is defined by $I=\dfrac{P}{A}$.

1. a) State the meaning of each symbol and the SI unit of irradiance.
2. b) A solar panel of area $1.6\ \text{m}^2$ receives radiation of power $1,200\ \text{W}$. Calculate the irradiance.
3. c) Explain what happens to the irradiance on a surface if the area is halved while the power is unchanged.

On a clear day in Cork the irradiance of sunlight on a horizontal surface is $800\ \text{W m}^{-2}$.

1. a) Calculate the power incident on a roof surface of area $25\ \text{m}^2$.
2. b) Calculate the energy incident on this surface in $2.0$ hours.
3. c) Express your answer to part b) in $\text{kWh}$.

Name the region of the electromagnetic spectrum associated with each of the following sources.

1. a) A microwave oven in a kitchen in Drogheda.
2. b) A TV remote control.
3. c) A dental X-ray machine.
4. d) A tanning lamp.
5. e) Which of these has photons of the greatest energy?

A point source of light, a lamp, emits radiation uniformly in all directions with a total power of $60\ \text{W}$.

1. a) Calculate the irradiance at a distance of $1.0\ \text{m}$ from the lamp, using the surface area of a sphere $A=4\pi r^2$.
2. b) Calculate the irradiance at $3.0\ \text{m}$.
3. c) State the relationship between irradiance and distance for a point source, and explain it with reference to the spreading of energy.

A solar farm in County Wexford has panels of total area $4,000\ \text{m}^2$. The average irradiance over a year is $120\ \text{W m}^{-2}$ and the panels are $18\%$ efficient.

1. a) Calculate the average power incident on the panels.
2. b) Calculate the average electrical power produced.
3. c) Calculate the electrical energy produced in one year, in $\text{kWh}$.
4. d) An average Irish household uses about $4,200\ \text{kWh}$ per year. Estimate how many households the farm could supply.

A student measures the irradiance of a small bulb at different distances using a light meter in a darkened room in Maynooth.

Distance $d$ / m	Irradiance $I$ / $\text{W m}^{-2}$
0.50	8.0
1.00	2.0
2.00	0.50

1. a) Show that the data are consistent with an inverse-square law.
2. b) Calculate the power of the bulb that is emitted as radiation, assuming it emits uniformly in all directions.
3. c) State one reason why the real value might differ from this calculation.

The Sun has a radiated power of $3.8\times10^{26}\ \text{W}$. The mean Earth–Sun distance is $1.5\times10^{11}\ \text{m}$ and the radius of the Earth is $6.4\times10^{6}\ \text{m}$.

1. a) Calculate the irradiance of sunlight at the top of the Earth's atmosphere (the solar constant).
2. b) Calculate the total power intercepted by the Earth, treating it as a disc of radius $6.4\times10^{6}\ \text{m}$.
3. c) Explain why the average irradiance over the Earth's whole surface is one quarter of the value in part a).
4. d) Explain why the irradiance on the ground in Ireland is lower than the value in part a), giving at least two reasons.

A flat solar water heater of area $3.0\ \text{m}^2$ is tilted so that sunlight of irradiance $900\ \text{W m}^{-2}$ strikes it at an angle of $40^\circ$ to the normal of the panel. The panel absorbs $85\%$ of the incident energy, and all of it heats water. Water flows through at a rate of $0.020\ \text{kg s}^{-1}$ with specific heat capacity $4,180\ \text{J kg}^{-1}\text{K}^{-1}$.

1. a) Explain why the effective irradiance on the panel is $I\cos\theta$, and calculate it.
2. b) Calculate the power absorbed by the panel.
3. c) Calculate the temperature rise of the water as it flows through the panel.
4. d) State the effect on the temperature rise of doubling the flow rate, and justify your answer.

A $5.0\ \text{mW}$ laser beam has a circular cross-section of diameter $2.0\ \text{mm}$. It is focused by a lens to a spot of diameter $20\ \mu\text{m}$.

1. a) Calculate the irradiance in the original beam.
2. b) Calculate the irradiance at the focused spot, assuming all the power passes through the lens.
3. c) Calculate the ratio of the two irradiances and explain the result.
4. d) Explain why looking into even a low-power laser beam can damage the eye, with reference to the focusing action of the eye's lens.