# Calculating B:I for Helmholtz coils

• Ryaners
In summary: I thought I was going mad. In summary, the conversation discusses the completion of an experiment measuring the dependence of magnetic field strength on current at the midpoint between two Helmholtz coils. The expected linear relationship was obtained from the data, but there is difficulty in calculating the result analytically. The necessary equations and measurements are provided, as well as Python code for the calculation. However, a mistake is discovered in the code where the radius of the coil is incorrectly inputted, leading to incorrect results.
Ryaners

## Homework Statement

I've completed an experiment where the dependence of magnetic field strength ##B## on current ##I## is measured at the midpoint along the axis between two Helmholtz coils (separation distance = coil radius ##r##). I got the expected linear relationship from the data but am having trouble calculating the result analytically.

##N## = number of turns per coil = 320
##\mu _0## = vacuum permittivity constant = ##4\pi \times 10^{-7}##
##r## = radius of coil = ##61.06 \times 10^{-3} m##

## Homework Equations

The ratio of ##B:I## at a point halfway along the axis between the coils (i.e. where ##x = \frac r 2##) is given by:
$$\frac {B}{I} = \left( \frac{4}{5}\right)^{\frac 3 2} \frac {\mu _0 N}{r}$$
This is derived as follows: the field from a single coil at distanct ##x## along the axis is:
$$B = \frac{\mu _0 N I r^2}{2 \left(x^2 + r^2\right)^{3/2} }$$
Adding a second coil in series with this doubles the field between them, and setting them at a separation of ##r## gives a uniform field. Setting ##x = \frac r 2##, i.e. at the midpoint, and dividing through by ##I## gives the expression above for ##\frac B I##.

The measurements were taking by positioning a probe at ##x = \frac r 2## and measuring ##B## for a range of ##I## values.

## The Attempt at a Solution

So from a graph of B against I, I get the expected result - about ##0.004 T A^{-1}##. BUT when I put the numbers into the above formula, I get something about 12 times bigger: ##0.048 T A^{-1}##. I can't see where I'm going wrong so I thought I'd see if someone here could shed some light on it. The derivation of the equation I'm using seems correct, & I'm not sure where a factor of 12 could be coming from - even if the number of turns in the coil is wrong, it would seem a bit excessive to be given coils with 3,840 turns for this experiment! (This is something I can confirm tomorrow when I get back to the lab.)

Any feedback is most welcome - thanks in advance.

If it makes things easier, here is some Python code for the above:

Python:
import matplotlib.pyplot as plt
import numpy as np

###
# Calculate the constant coefficient analytically.
###

# vacuum permittivity constant:
mu_0 = 4.0 * 10.0**(-7) * np.pi
# no. of turns each coil
N = 320.
# radius of coil ( = separation distance between coils)
r = 0.00601

#equation for ratio of B:I for Helmholtz coils with this spacing
analytic = ((4/5)**(3/2)) * mu_0 * N / r

print("Analytical Result: ", analytic)

###
# Experimental Data. Plot a graph, fit a line & print the slope.
###

# measurements:
I = np.array([0.,0.025,0.051,0.074,0.099,0.125,0.150,0.175,0.201,0.299,0.4,0.5,0.6,0.7,0.801,0.901,0.999])
B_mT = np.array([0.,0.11,0.21,0.3,0.42,0.5,0.61,0.69,0.81,1.19,1.6,1.9,2.3,2.8,3.22,3.63,4.03])

# convert B measurements from milliTesla to Tesla
B = B_mT * 10**(-3)

# linear regression
p = np.polyfit(I, B, 1)
m = p[0]
c = p[1]
line = m * I + c

print("Result from Graph: ", m)
print("Ratio of Analytical : Experimental Result = ", analytic / m)

xerr = 0.01
yerr = 0.00005

# plot graph
fig1 = plt.figure()
plt.grid(which='both', axis='both', linestyle='-')
# label axes, set axis limits
plt.xlabel('Current [A]')
plt.ylabel('Magnetic Field Strength [T]')
plt.xlim(0.0,1.05)
plt.ylim(0.0, 0.00425)
plt.errorbar(I, B, yerr, xerr, '.', capsize=3)
plt.title('Dependence of Magnetic Field on Current (in Helmholtz Coil)')
plt.plot(I, B, '.')
plt.plot(I, line, 'r')

plt.show()

Your code says r = 0.00601, I assume the number is in meters. That's 60.01×10-4m.
You say
Ryaners said:
r = radius of coil = 61.06×10−3m
Which is it? As we say in the trade, "Garbage in, garbage out."

kuruman said:
Your code says r = 0.00601, I assume the number is in meters. That's 60.01×10-4m.
You say

Which is it? As we say in the trade, "Garbage in, garbage out."

Ah, yes, that does make a quite a bit of a difference alright, whoops Thank you for pointing out the mistake! Fresh pair of eyes etc.

## 1. How do I calculate the magnetic field intensity (B) for Helmholtz coils?

To calculate the magnetic field intensity for Helmholtz coils, you will need to use the formula B = μ₀IN/2R, where μ₀ is the permeability of free space (4π x 10^-7), I is the current flowing through the coils, N is the number of turns in each coil, and R is the radius of the coils.

## 2. What units are used for B:I calculations for Helmholtz coils?

The units used for B:I calculations for Helmholtz coils are Tesla (T) or Gauss (G), depending on the preference of the scientist. It is important to be consistent with units throughout the calculation.

## 3. Can the B:I calculation be used for any size of Helmholtz coils?

Yes, the B:I calculation can be used for any size of Helmholtz coils as long as the necessary parameters (μ₀, I, N, R) are known. However, it is important to note that the formula assumes that the coils are perfectly aligned and have a uniform current distribution.

## 4. How can I verify the accuracy of my B:I calculation for Helmholtz coils?

To verify the accuracy of your B:I calculation, you can use a gaussmeter to measure the magnetic field intensity at different points along the axis of the coils. Compare these measurements to your calculated B:I value and make adjustments as needed.

## 5. Are there any safety precautions to consider when working with Helmholtz coils and calculating B:I?

Yes, it is important to take necessary safety precautions when working with Helmholtz coils and calculating B:I. Always ensure that the power supply is turned off before making any adjustments to the coils. Also, be aware of the potential hazards of strong magnetic fields and take appropriate measures to protect yourself and others in the vicinity.

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