Derivation of total magnetic field with changing distance in a coil?

In summary, the magnetic field along the axis of a coil of wire can be calculated using the Biel-Savart Law and the equation for the current in the wire. The magnitude of the magnetic field can be calculated as a function of the position along the axis, the number of turns of wire, and the electric current in the coil.
  • #1
Violagirl
114
0

Homework Statement



You are a member of a research team studying magnetotactic bacteria. Magnetotactic
bacteria from the southern hemisphere preferentially swim to the south along magnetic
field lines, while similar bacteria from the northern hemisphere preferentially swim to
the north along magnetic field lines. Your team wishes to quantify the behavior o f
magnetotactic bacteria in closely controlled magnetic fields.

You know from your physics class that a coil of wire can be used to produce a magnetic
field, which can be varied by changing the current through it. You set yourself the task
of calculating the magnetic field along the axis of the coil as a function of its current,
number of turns, radius, and the distance along the axis from the center of the coil. To
make sure you are correct, you decide to compare your calculation to measurements.

Calculate the magnitude of the magnetic field as a function of the position along the
central axis of a coil of known radius, the number of turns of wire, and the electric
current in the coil.

Homework Equations


Biel-Savart Law: B = µ0/4π ∫c Idl x r / r3

The Attempt at a Solution



I was going to see if someone would be able to check to see if I did the derivation correctly? Thanks so much!

See attached document for diagram of situation.

Derived equation:
∫ dBz = ∫ µ0 N I dl cos θ/4π (R2 + z2) (R/√R2 + z2)

µ0 N I R / 4π (R2 + z2)3/2 ∫ dl

2π µ0 I N R2 / 4π (R2 + z2)3/2

µ0 I N R2 / 2 (R2 + z2)3/2-final equation.
 

Attachments

  • Vector alignment of magnetic field of coil.docx
    21.5 KB · Views: 238
Physics news on Phys.org
  • #2
Man, that's really messy...
Reprint of the work, as is.
##\vec{B} = \frac{\mu_0}{4\pi}\int_C \frac{I d\vec{\ell}\times\vec{r}}{|r|^3}## -> note that ##\frac{\vec{r}}{|r|} = \hat{r}##
##\int dB_z = \int \mu_0 N I \frac{d\ell \text{cos}(\theta)}{4\pi (R^2 + z^2)}\frac{R}{R^2 + z^2}##
... to be edited later...
 
  • #3
BiGyElLoWhAt said:
Man, that's really messy...
Reprint of the work, as is.
##\vec{B} = \frac{\mu_0}{4\pi}\int_C \frac{I d\vec{\ell}\times\vec{r}}{|r|^3}## -> note that ##\frac{\vec{r}}{|r|} = \hat{r}##
##\int dB_z = \int \mu_0 N I \frac{d\ell \text{cos}(\theta)}{4\pi (R^2 + z^2)}\frac{R}{R^2 + z^2}##
... to be edited later...
Continuing because it's too late to edit.
***
##\frac{\mu_0 NIR}{4\pi (R^2 + z^2)^{3/2}}\int_l d \ell## This is good, assuming that the point only lies on the z axis, about which the loop is centered (the z axis is normal to the area of the loop) and also that the wires have negligible width (all the loops are concentrated in a small range of z, call it A, such that A<<z). The limits (l) of the integral should be 1 full circle, so for example, 0 to 2pi r.
##\frac{\mu_0 NIR}{4\pi (R^2 + z^2)^{3/2}}(2\pi R)##
##\frac{\mu_0 NIR}{2 (R^2 + z^2)^{3/2}}## Given these assumptions, this is a valid solution.
 
  • #4
Originally, I wasn't thinking that this was simplified as such (see the assumptions). I'm working on a general form, lacking these assumptions, that would work for any point in R^3. However, I think I might have to essentially treat the coil as a cylindrical shell with current density J, and it keeps getting hairier and hairier. =D

I'll post back... eventually.
 
  • #5
BiGyElLoWhAt said:
Originally, I wasn't thinking that this was simplified as such (see the assumptions). I'm working on a general form, lacking these assumptions, that would work for any point in R^3. However, I think I might have to essentially treat the coil as a cylindrical shell with current density J, and it keeps getting hairier and hairier. =D

I'll post back... eventually.
You will encounter hopelessly complex math, e.g. elliptic integrals plus the need to consider the radius of the wire.
Quit now while the getting is good! :smile:
 
  • #6
=D
Yea that's what I was finding. I actually looked up the solution to one of the integrals, and it was about 5 ridiculous substitutions... I still kind of want to work it out, but we'll see how it gets. Good looking out ;)
 
  • #7
BiGyElLoWhAt said:
=D
Yea that's what I was finding. I actually looked up the solution to one of the integrals, and it was about 5 ridiculous substitutions... I still kind of want to work it out, but we'll see how it gets. Good looking out ;)
Yes, and even then it will be an approximation to an arbitrary order.
 
  • Like
Likes BiGyElLoWhAt

1. How is the total magnetic field in a coil affected by changes in distance?

The total magnetic field in a coil is directly affected by changes in distance. As the distance from the coil increases, the strength of the magnetic field decreases. This is because the magnetic field follows an inverse square law, meaning that the strength of the field is inversely proportional to the square of the distance from the source.

2. What is the equation for calculating the total magnetic field in a coil at a given distance?

The equation for calculating the total magnetic field in a coil at a given distance is B = μ₀N/l, where B is the magnetic field strength, μ₀ is the permeability of free space, N is the number of turns in the coil, and l is the length of the coil.

3. Can the total magnetic field in a coil be increased by increasing the number of turns?

Yes, the total magnetic field in a coil can be increased by increasing the number of turns. This is because each turn of the coil adds to the overall magnetic field strength. However, this increase is not linear and follows the same inverse square law as changes in distance.

4. How does the diameter of the coil affect the total magnetic field?

The diameter of the coil also affects the total magnetic field. A larger diameter coil will have a weaker magnetic field than a smaller diameter coil with the same number of turns. This is because the magnetic field strength is spread out over a larger area in a larger coil, while in a smaller coil it is more concentrated.

5. Is there a limit to how far away the magnetic field of a coil can be detected?

Yes, there is a limit to how far away the magnetic field of a coil can be detected. As the distance increases, the magnetic field becomes weaker and can eventually become undetectable. This distance is dependent on the strength of the magnetic field and the sensitivity of the detector being used.

Similar threads

  • Introductory Physics Homework Help
Replies
15
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
208
  • Introductory Physics Homework Help
Replies
4
Views
269
  • Introductory Physics Homework Help
Replies
12
Views
2K
Replies
49
Views
3K
  • Introductory Physics Homework Help
Replies
12
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
664
  • Introductory Physics Homework Help
Replies
7
Views
783
  • Introductory Physics Homework Help
Replies
25
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
2K
Back
Top