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Homework Help: Energy, magnitude of E & B, pressure from a laser

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    A Ti-sapphire femtosecond laser (1fs= 10^-15s, λ≈0.8μm) has an intensity of 1022 W/cm2
    when focused to a spot of 1μm radius. What is the energy of the laser pulse? What is the peak
    magnetic induction B in Tesla and the electric field in V/m? What is the wave pressure in
    atmospheres?

    2. Relevant equations

    [tex]P = \int I da = \frac{E}{\Delta t}[/tex]
    [tex]E=cB[/tex]

    3. The attempt at a solution

    I've solved the first part
    [tex]P = IA = 10^{22} \frac{W}{cm^2} \pi \left(1 \mu m\right)^2 [/tex]
    [tex] = 3.14 \times 10^{14} W[/tex]
    [tex]E = .314 J [/tex]

    But what is the relationship of what was given to either ]E or B? The only equations I can find are sinusoidal with the assumption that [tex]E_0[/tex] or [tex]B_0[/tex] are known to find their values at a particular time. Would I take the pulse with the assumption that B and E = 0 and the beginning and end and the information of the wavelength to find the value it peaks at? Now that I'm thinking of that, I'd still need some sort of E_0 or B_0.
     
    Last edited: Apr 27, 2010
  2. jcsd
  3. Apr 27, 2010 #2
    Assuming you have done the first part correct (if the question suggests it, you may need to integrate a gaussian distibution to get the power instead of just using the area of a circle)... anyway, to answer your question. The irrandiance [itex]I[/itex] relates to both the electric field [itex]E[/itex] and the magnetic induction [itex]B[/itex] through its definition as the time averaged Poynting vector:

    [tex]
    I \equiv \left\langle S \right\rangle = \frac{c^2\epsilon_0}{2}\left\vert E_0 \times B_0 \right\vert
    [/tex]

    So that:
    [tex]
    I = \frac{c}{2\mu_0}B_0^2
    [/tex]
    [tex]
    I = \frac{1}{2}c\epsilon_0 E_0^2
    [/tex]

    The "average" radiation pressure is given by the energy density of the wave (note the units energy/volume and force/area are the same)

    [tex]P = \frac{1}{2}\epsilon_0 E_0^2[/tex] (this will give answer in Pascals though using previous units)

    Hope this helps...
     
  4. Apr 27, 2010 #3
    That does help immensely thank you.
     
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