Start a Fire with Shaving Mirror?

[Master J]

If paper ignites at 500 C, can one start a fire with a typical shaving mirror using sun light?


Since the only time a concave mirror has an upright, virtual, magnified image is with the object between the mirror and focus, I estimate a focal length of about 1m.

Now, the sun is at infinity, so all its light is passed thru the focus. This is where I get stuck. How do I relate the magnification of the mirror to the increase in intensity. I could then use Stefan- Boltzmann to get an idea of temperature, but I am stuck here. Any ideas?


Thanks!

 

[chaoseverlasting]

If they're saying that a shaving mirror is to be used, its a plain mirror with infinite focal length.

What you have to do is calculate the energy received per unit area from the sun. Now, the let the energy recived per unit area be $$l$$ (I forget the symbol usually used for this constant).

Let the surface area of the mirror be $$A$$.

We'll assume that all of the energy falling on the mirror falls on the paper normally.

Let the ambient temperature be $$T_0$$ and the temperature of the paper at any time t be $$T$$.

As the paper is not a perfect black body, the law is of the form:

$$P=\sigma \epsilon AT^4$$

where P represents the power absorbed by the paper.
As the paper will also radiate energy to its surroundings, our law takes the form

$$P=\sigma \epsilon A(T-T_0)^4$$

Power is energy per unit time or $$P=\frac{dE}{dt}$$.

This is when I'm a little shaky on the subject. It's been a long time since I've done these problems so I may make a mistake.

If I remember correctly, the heat energy required to raise the temperature by a value $$\delta T$$ is given by:

$$q=mc\delta T$$

Where m is the mass of the paper and C is a thermodynamic constant.

From the earlier equation we know,

$$P=\frac{dE}{dt}$$

Simplifying the above experssion we have E=P.t

Equate this to q and you should be able to get a result.

Please double check my work as I may have made a mistake. There are many, more capable posters on this forum and I'm a little out of touch.

 

[Master J]

Thanks for the input...I'll go thru your solution shortly.However, the question says "typical concave shaving mirror". I thought that was usually the case since one would normally like to magnify the image?

I should add that no figures are given for the question except the temperature at which paper ignites.

 

[chaoseverlasting]

Thanks for the input...I'll go thru your solution shortly.


However, the question says "typical concave shaving mirror". I thought that was usually the case since one would normally like to magnify the image?

I should add that no figures are given for the question except the temperature at which paper ignites.

In that case, all the energy is focused on a very small area. At the focal point of the mirror, the energy/unit area will be maximum and the paper will ignite after a time.