A microwave operates at 250 . W. Assuming that the waves...

A microwave operates at $250 .$ W. Assuming that the waves emerge from a point source emitter on one side of the oven, how long does it take to melt an ice cube $2.00 \mathrm{~cm}$ on a side that is $10.0 \mathrm{~cm}$ away from the emitter if $10.0 \%$ of the photons that strike the cube are absorbed by it? How many photons of wavelength $10.0 \mathrm{~cm}$ hit the ice cube per second? Assume a cube density of $0.960 \mathrm{~g} / \mathrm{cm}^{3}$

Key Concepts

Microwave Power and Energy Transfer

This concept involves understanding that power is the rate at which energy is delivered. In the context of the problem, a microwave operating at a specified wattage delivers energy over time that is used to induce physical changes (like melting ice). Calculating the time to effect a change involves multiplying the power by the time interval and equating it to the energy needed for the process.

Latent Heat and Phase Change

Latent heat is the energy required to change the phase of a substance without changing its temperature. For melting an ice cube, the latent heat of fusion must be overcome. The calculation requires knowing the mass of the ice and the specific latent heat of fusion for ice, which together provide the total energy required to cause the phase transition from solid to liquid.

Photon Energy and Electromagnetic Radiation

Electromagnetic radiation is quantized in the form of photons, and each photon’s energy is given by the equation E = hc/?, where h is Planck’s constant, c is the speed of light, and ? is the wavelength. This concept is essential when determining how many photons are needed to supply a particular amount of energy, such as the energy necessary to melt the ice.

Spatial Distribution and the Inverse Square Law

When radiation emanates from a point source, its intensity decreases with the square of the distance from the source. This inverse square law is important to understand because it describes how the microwave energy spreads out over an area, which in turn affects the amount of energy that reaches the ice cube based on its distance from the emitter.

Absorption Efficiency

Absorption efficiency refers to the fraction of incident radiation that is actually absorbed by a target material. In this problem, only a certain percentage of the photons that strike the ice cube contribute to the energy transfer needed to melt the ice. Understanding this concept helps in accurately computing the effective energy delivered to the ice cube.

Transcript

00:01 For this problem on the topic of electromagnetic waves, we're told that a microwave is operating at 250 watts of power.

00:07 Now we want to assume that the waves emerge from a point source emitter on one side of the oven, and we want to know how long it will take for an ice cube that has a side length of 2 centimeters to melt if it is 10 centimeters away from the emitter and 10 % of the photons are absorbed by the cube.

00:24 We want to know how many photons of wavelength 10 centimeters will hit the ice cube per second, and we're given the density of the ice cube to be 0 .96 grams per cubic centimeter.

00:35 Now the energy required to melt the ice is given by the latent heat of fusion for ice cf, which is 334 joules per gram.

00:48 The intensity of light at the cube, i is equal to the radiation power p -0 over the area of the beam 4 pi d squared.

01:02 And the radiation power that's incident on the cube is il squared.

01:07 So the power absorbed by the cube p is equal to epsilon i .l squared, which is the energy incident on the cube over time.

01:20 Now the mass of ice, m is given by row l cubed, which is its density row times its volume v, and the energy of one photon will call it e, p .h.

01:32 Is equal to hc over lambda or hf.

01:38 And so the energy required to melt the ice, we'll call it em, is equal to the mass of ice times the latent heat of fusion.

01:49 And this is equal to the density of the ice cubed times its volume, l cubed, times the latent heat of fusion, cf.

01:59 And so the power absorbed by the ice cube p is equal to epsilon times i times l squared, which is epsilon times l squared.

02:13 And the intensity is the radiation power, p -not, over the area 4 pi d squared.

02:22 And we know that this is equal to the energy incident over time t...

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