- #1

Ryaners

- 50

- 2

## 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()
```