Helmholtz Coil Problem: Analyzing Magnetic Field at Point Z

In summary, the Helmholtz coil consists of two identical circular coils of wire, each with N loops and carrying current I, placed R meters apart. The magnetic field at point Z, midway between the coils, is directed to the left and its strength can be expressed as a function of the variable z. The first and second derivatives of the magnetic field strength with respect to z are both zero at point Z, indicating that the field varies very little at this point. To find the magnetic field at any point on the z axis, one can use the equation dB_z=(mu_0I/4pi)(dl/r^2)cos(theta) and integrate the contributions from all the dl elements of the coils, multiplied by the factor
  • #1
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Homework Statement



The diagram illustrates a Helmholtz coil, consisting of two identical circular coils of wire, each one having N loops and carrying the same current I. The radius of each coil is R, and the two coils are R meters apart. (a) What are the direction and strength of the magnetic field at the point Z, midway between the two coils? (b) Express B as a function of the variable z, taking the origin for z to be at the center of the left-hand coil and the positive z direction to be to the left. (c) Find the first and second derivatives of the magnetic field strength B with respect to z, dB/dz and d 2B/dz2. Show that they are both zero at point Z (where z = −R/2). This shows that the magnetic field varies very little at the point Z.

http://i40.tinypic.com/k4b5ag.jpg

that link should have the picture.

Homework Equations



I think the equation for a Helmholtz coil is (4/5)3/2 (uNI/R)


The Attempt at a Solution



For part a, I think it's to the left. I could really use some direction for parts B and C. Thanks!
 
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  • #2
Start by finding the differential field at any point on the z axis for one current loop:

[tex]dB_z=\frac{\mu_0I}{4\pi}\frac{dl}{r^2}cos\theta[/tex]

where theta is the angle made by the radius of the loop and the line from the current element on the loop to the point on the z axis. Intergrate to determine the field contributions from all the dl elements. Then multiply by the factor N to find all the loop contributions. Do the same for the second coil. Express the cosine function in terms of the loop radius and the distance from one of the loop centers to the point in question. Finish by using superposition.
 

1. What is a Helmholtz coil?

A Helmholtz coil is a type of electromagnetic coil consisting of two identical circular coils placed in parallel, with their axes aligned and a distance equal to their radius. This configuration produces a nearly uniform magnetic field in the space between the coils.

2. What is the purpose of analyzing the magnetic field at point Z?

Analyzing the magnetic field at point Z allows scientists to understand the strength and direction of the magnetic field produced by the Helmholtz coil. This information can be used for various applications such as calibrating magnetic sensors or studying the behavior of charged particles in a magnetic field.

3. How is the magnetic field at point Z calculated for a Helmholtz coil?

The magnetic field at point Z can be calculated using the equation B = μ₀NI/R, where B is the magnetic field strength, μ₀ is the permeability of free space, N is the number of turns in the coil, I is the current flowing through the coil, and R is the distance from the center of the coil to point Z.

4. What factors can affect the magnetic field at point Z in a Helmholtz coil?

The magnetic field at point Z can be affected by the distance from the center of the coil, the number of turns in the coil, and the current flowing through the coil. Additionally, external magnetic fields or nearby conductive materials can also alter the strength and direction of the magnetic field.

5. What are some practical applications of Helmholtz coils?

Helmholtz coils have various practical applications, such as producing a uniform magnetic field for medical imaging devices, calibrating magnetic sensors and compasses, and conducting experiments in physics and engineering. They are also used in research and development for studying the behavior of particles in a magnetic field.

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