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I've been studying the attached paper - "A Pulsed Bomb Locator" by F.B. Johnson M.A.
Equation 1 (at the bottom of page 2):
^{\frac{3}{2}}}\ \ [oersted])
This represents the magnetic field at a buried spherical target.
In MKS units (from John Alldred's comments):
^{\frac{3}{2}}}\ \ [Am^{-1}])
OK - this is all very well, but what is the origin of this equation?
From the Biot-Savart Law:
i.e. The field of dB due to a small element dl of the circle, centered at A (a point on the circumference of the coil) is given by:
where:
x = distance along the coil axis
r = radius of coil
dl = infinitesimal length of conductor carrying current I
= unit vector specifying distance to the field point
= permeability of free space
There are two components to the field. One is along the coil axis, and the other is perpendicular to the axis. Hence, the total magnetic field at a distance along the coil axis is:
^{\frac{3}{2}})
which now starts to bear some resemblance to equation 1.
If there are T turns on the coil, then:
^{\frac{3}{2}})
and (since
we have:
^{\frac{3}{2}}}\ \ [Am^{-1}])
which is John Alldred's MKS version of equation 1.
So far, so good ... but we then move on to equation 2:
where:
I = the intensity of magnetization of a sphere (target object)
= decay time of eddy currents in the target
which is based on the assumption that the switch-off is less than one-tenth of the decay time of eddy currents in the target object.
Despite repeated attempts, I am failing to derive this formula, although it appears that the volume of a sphere
is in there somewhere.
And then we have equation 3:
where:
a = radius of target sphere [cms]
= conductivity of material [emu]
John Alldred converted this to MKS units as follows:
I'm not certain, but equation 3 seems somewhat suspect to me.
For example ... let's take a small sphere of pure gold with a diameter of 5mm. In this case we have:
a = 2.5 x
= 4.10 x 
This yields (from the MKS version of equation 3) a of 32.6us ... which seems too high to me.
If anyone fancies some homework, then see if you can derive equation 2, and figure out what's wrong with equation 3.
I did think this might be useful to compare with the work that Green has been doing in determining the time constants of some standard metal targets, but now I'm confused.
Equation 1 (at the bottom of page 2):
This represents the magnetic field at a buried spherical target.
In MKS units (from John Alldred's comments):
OK - this is all very well, but what is the origin of this equation?
From the Biot-Savart Law:
i.e. The field of dB due to a small element dl of the circle, centered at A (a point on the circumference of the coil) is given by:
where:
x = distance along the coil axis
r = radius of coil
dl = infinitesimal length of conductor carrying current I
There are two components to the field. One is along the coil axis, and the other is perpendicular to the axis. Hence, the total magnetic field at a distance along the coil axis is:
which now starts to bear some resemblance to equation 1.
If there are T turns on the coil, then:
and (since
which is John Alldred's MKS version of equation 1.
So far, so good ... but we then move on to equation 2:
where:
I = the intensity of magnetization of a sphere (target object)
which is based on the assumption that the switch-off is less than one-tenth of the decay time of eddy currents in the target object.
Despite repeated attempts, I am failing to derive this formula, although it appears that the volume of a sphere
And then we have equation 3:
where:
a = radius of target sphere [cms]
John Alldred converted this to MKS units as follows:
I'm not certain, but equation 3 seems somewhat suspect to me.
For example ... let's take a small sphere of pure gold with a diameter of 5mm. In this case we have:
a = 2.5 x
This yields (from the MKS version of equation 3) a
If anyone fancies some homework, then see if you can derive equation 2, and figure out what's wrong with equation 3.
I did think this might be useful to compare with the work that Green has been doing in determining the time constants of some standard metal targets, but now I'm confused.
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