Creating a Zero-Field Zone with an Atom Chip

[tim_3491]

Homework Statement

Show that if a straight wire carrying current I is oriented at right angles to a uniform external magnetic field B, there will be a region where the magnetic field is zero. If the wire is actually ‘printed’ onto the surface of an ‘atom chip’ and the atoms are to be trapped 500 micrometre from the surface using an external field of 25 x 10^-4 T, what current will the wire need to carry? If the resistance of the wire, which is 6mm long and 150µm wide is not to exceed 0.1 ohm, how thick must it be if made of copper?
(See http://www.physics.uq.edu.au/atomoptics/atomchip.html for the actual device we made.
The potential energy of atoms in one spin state is zero where the field is zero, so they can be trapped there.

Homework Equations

Anyone have any ideas, we have recently been doing emf and electric fields

The Attempt at a Solution

Not to sure even on how to start it

 

[anathema]

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Thank you for your post. I would be happy to help you with your problem. First, let's review the concepts of magnetic fields and currents.

A magnetic field is a region of space where a magnetic force can be felt. It is created by moving charges, such as the electrons in a wire carrying current. The strength of the magnetic field is measured in units of Tesla (T).

A current is the flow of electric charge, usually in the form of electrons, through a conducting material. The amount of current is measured in units of Amperes (A).

Now, to your problem. We are dealing with a straight wire carrying current I, which is oriented at right angles to a uniform external magnetic field B. When a current-carrying wire is placed in a magnetic field, a force is exerted on the wire. This force is given by the equation F = I * L * B, where I is the current, L is the length of the wire, and B is the magnetic field. Since the wire is oriented at right angles to the magnetic field, the force will be perpendicular to both the current and the magnetic field.

Now, let's consider the region where the magnetic field is zero. This will occur when the force on the wire is also zero. From the equation above, we can see that this will happen when either the current or the length of the wire is zero. Since we cannot have a wire with zero length, the only way for the force to be zero is for the current to be zero. This means that there will be a region along the wire where the magnetic field is zero.

Next, let's determine the current needed for the wire to carry in order to trap atoms 500 micrometers from the surface using an external magnetic field of 25 x 10^-4 T. We can use the equation F = I * L * B again, but this time we will solve for I. Rearranging the equation, we get I = F / (L * B). Plugging in the values given, we get I = (25 x 10^-4 T) / (500 x 10^-6 m) = 0.05 A. Therefore, the wire will need to carry a current of 0.05 Amperes in order to trap the atoms 500 micrometers from the surface.

Finally, we need to determine the required thickness of the wire if

 

[Moetasim]

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I can say that the concept of creating a zero-field zone with an atom chip is a fascinating and innovative idea. The use of a straight wire carrying current at a right angle to an external magnetic field is a well-known phenomenon in electromagnetism. This setup creates a region where the magnetic field is zero, which can be utilized to trap atoms in their lowest energy state.

To determine the current needed for this setup, we can use the equation B = μ0I/2πr, where B is the magnetic field, μ0 is the permeability of free space, I is the current, and r is the distance from the wire. In this case, we have B = 25 x 10^-4 T, r = 500 micrometers (or 0.5 mm), and μ0 = 4π x 10^-7 Tm/A. Solving for I, we get I = 2πrB/μ0 = 3.14 x 0.5 x 25 x 10^-4/4π x 10^-7 = 1.96 x 10^-3 A.

Next, we need to determine the thickness of the wire to ensure that the resistance does not exceed 0.1 ohm. We can use the equation R = ρl/A, where R is the resistance, ρ is the resistivity of the material, l is the length of the wire, and A is the cross-sectional area. In this case, we have R = 0.1 ohm, ρ = 1.68 x 10^-8 ohm*m (resistivity of copper), l = 6 mm, and A = x x 150 x 10^-6 m^2 (where x is the thickness of the wire). Solving for x, we get x = ρl/R = 1.68 x 10^-8 x 6 x 10^-3/0.1 = 1.008 x 10^-4 m = 100.8 µm.

Therefore, the wire must have a thickness of approximately 100.8 µm to ensure that the resistance does not exceed 0.1 ohm. This is a feasible thickness for a wire to be printed onto the surface of an atom chip.

In conclusion, creating a zero-field zone with an atom chip is a promising technique for trapping atoms