1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability that a magnetic dipole is oriented with theta

  1. Jan 28, 2016 #1
    1.
    the problem goes like this :
    The energy of interaction of a classical magnetic dipole with the magnetic field B is given by
    E = −μ·B.
    The sum over microstates becomes an integral over all directions of μ. The direction of μ
    in three dimensions is given by the angles θ and φ of a spherical coordinate system
    The integral is over the solid angle element
    = sin θdθdφ. In this coordinate system
    the energy of the dipole is given by E = −μB cos θ.
    Choose spherical coordinates and show that the probability p(θ, φ)dθdφ that the dipole is
    between the angles θ and + dθ and φ and φ + dφ is given by

    p(θ, φ)dθdφ = (e^(μB cos θ) sin(θ) dθ dφ)/z



    2. Relevant equations


    z = ∫∫ e^(μB cos θ) sin(θ) dθ dφ .


    3. The attempt at a solution
    i have no idea what to do , and i tried all i know
    i know that the boltzmann distribution gives you the probability that a particle has an energy is :
    e^(
    μB cos θ)/∫e^(μB cos θ) , but how do i integrate the spherical coordinates i don t know . please help me and thank you .
     
    Last edited: Jan 28, 2016
  2. jcsd
  3. Jan 28, 2016 #2
    this is the best that i could do
    the probability that the dipole between x and dx is
    dp(x) = (1 / Z ) e^(μ(x)B cos θ) dx = dx because we assume that B is parallel to z so μ(x) . B = 0
    dp(y) =
    (1 / Z ) e^(μ(y)B cos θ) dy = dy
    dp(z) = (1 / Z )e^(μ(z)B cos θ) dz
    d^3 p( x,y,z)= (1 / Z )e^(μ(z)B cos ) dz dy dx = (1 / Z )[B][B][B][B]e^(μ(z)B cos [B][B][B][B]θ[/B][/B][/B][/B]) dv[/B][/B][/B][/B]
    d^3 p( r,
    θ,[B][B][B][B])[/B][/B][/B][/B]
    = (1 / Z )e^(μ(z)B cos θ) rd²r d[B][B][B][B]θ dφ[/B][/B][/B][/B]

    is this true ? or am i making horrible mistakes ?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Probability that a magnetic dipole is oriented with theta
  1. Magnetic Dipole (Replies: 1)

Loading...