Probability that a magnetic dipole is oriented with theta

In summary, the problem involves finding the probability of a dipole being between certain angles in a spherical coordinate system. The energy of the dipole is given by E = -μB cos θ and the probability is given by p(θ, φ)dθdφ = (e^(μB cos θ) sin(θ) dθ dφ)/z. The integral for z is z = ∫∫ e^(μB cos θ) sin(θ) dθ dφ. The Boltzmann distribution is used to calculate the probability and the dipole is assumed to be parallel to the z-axis. The final expression for the probability is d^3 p( r,θ,)= (1
  • #1
potatowhisperer
31
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. Homework Equations


z = ∫∫ e^(μB cos θ) sin(θ) dθ dφ .

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^([/B]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 .
 
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  • #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 )e^(μ(z)B cos θ) dv
d^3 p( r,
θ,)
= (1 / Z )e^(μ(z)B cos θ) rd²r dθ dφ

is this true ? or am i making horrible mistakes ?
 

FAQ: Probability that a magnetic dipole is oriented with theta

1. What is the concept of "magnetic dipole orientation"?

The magnetic dipole orientation refers to the direction in which a magnetic dipole is pointing. This is determined by the orientation of the north and south poles of the dipole.

2. How is the magnetic dipole orientation measured?

The magnetic dipole orientation is typically measured using a magnetic dipole moment, which is a vector quantity that represents the strength and direction of the dipole. This measurement can also be expressed in terms of the angle theta, which represents the angle between the dipole and a fixed reference direction.

3. What is the significance of the probability of a magnetic dipole being oriented with theta?

The probability of a magnetic dipole being oriented with theta is important in understanding the behavior of magnetic materials. It can help predict the strength and direction of the magnetic field produced by the dipole, as well as how it will interact with other magnetic materials.

4. How does temperature affect the probability of a magnetic dipole being oriented with theta?

At higher temperatures, thermal energy can cause the magnetic dipoles to become more randomly oriented, leading to a decrease in the probability of a specific theta value. At lower temperatures, the dipoles are more likely to align with a specific theta, resulting in a higher probability.

5. Can the probability of a magnetic dipole being oriented with theta be manipulated?

Yes, the probability of a magnetic dipole being oriented with theta can be manipulated by applying an external magnetic field or by changing the temperature. Additionally, certain materials can be engineered to have a higher probability of a specific theta value, making them useful for various applications.

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