First Homework Problem Set - Engineering 1931F Spring 2015
1.) An MRI machine is one of the few examples of storing electrical energy on some
significant scale. The most advanced machines use a magnetic field of about 3.0T and
have a working volume on the inside of the magnet around 2' dia x 3' long.
1.1) Find the stored magnetic energy per cubic meter at 3.0T and the total stored energy
in the working volume of the MRI. (Of course, the refrigeration of the superconducting
magnet uses substantial power itself, making this a very inefficient means of storage.)
1.2) The United States produces about 1.5 KW of electrical power per person at all times.
How many person-hours of energy is stored in the MRI?
2.) For the magnetic circuit below, find the magnitude of B in the air gap and in the
center of the coil. Also find the flux,  , in the gap and the numerical value of the
reluctance of the gap in SI units.
3.) A donut-shaped, soft iron ring has a circular cross-section 3 cm in diameter and a
mean circumference of 80 cm. (You may assume the B and H fields are uniform across
the cross section.) The ring is uniformly wound with a coil of 600 turns. The graph
below shows the DC magnetization curve for the material. Estimate the current required
to produce a flux of 5.7 104 webers in the ring. (The weber is the SI unit for flux and is
a total of 1 square meter at 1 Tesla.) If a saw cut creates a 2 mm air gap in the ring, find
the approximate flux in the gap produced by the same current. Find the current which
will produce the same flux in the gap as before the saw cut. Neglect fringing and leakage
of the field around the structure.
ENGN1931F – Spring 2015
Homework 1
4.) The magnetic circuit below is made of the same material as the last problem. Find the
current needed to create a 10,000 gauss field in the gap. Under that condition what is the
energy stored in the coil, in the air gap, and in the iron? What is the (very) approximate
N M
will do for this.)
inductance of the coil? ( L 
I
5.) You saw the little solenoid for a washing machine that I used to motivate doing
problems of this kind. I am addicted to reverse engineering so one thing I thought of but
have not done well is to figure out the number of turns to the coil on that unit. I would
not want to damage the coil in any way. Describe an experiment you could do to
determine that value. The frame of the solenoid has a constant cross section of 1 x 1 cm
and a magnetic path length of about 10 cm. Estimating the iron permeability is tricky. At
the relatively high current and voltage of its intended operation the permeability is
probably around 5000 based on what typical solenoid iron is supposed to have. Used at
low current for lab measurements the effective permeability will be much lower, maybe
as low as 1000. What equations would you use? How would you set it up to make the
measurement easy and as independent as possible of the iron properties? What would you
measure?
6.) The storage of energy, particularly electrical energy, is extremely difficult to do well.
In the first problem of this set, you calculated the energy in a cubic meter of high
magnetic field – not a very practical method but still it’s a number. I would like you to
compare that density to the energy density of other storage methods:
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ENGN1931F – Spring 2015
Homework 1
6.1) Plastics can easily withstand 100,000 volts per centimeter and some reasonably
priced ones have dielectric constants of around 5. What is the energy stored in a cubic
meter of such material?
6.2) My home is heated with #3 heating oil and the company that sells it to me says it
stores 115,000 BTU per gallon. How much is stored in a cubic meter of this oil?
6.3) Pumped water storage is often proposed as a way to use daytime generated power at
night. Suppose Nature was kind enough to give you two lakes about 1000 feet different
in elevation and near enough to be connected with a practical pipe. How much energy
would you store by pumping a cubic meter of water up to the top?
6.4) Hydrogen is sometimes thought of as a way to power cars. To make the comparison
unfair, calculate how much energy is stored in a cubic meter of pure hydrogen at NTP.
6.5) One of the groups in my ENGN1650 class is using a lithium battery that is a pretty
good example of the capacity of small batteries of that type. The battery is an Ultralife
UBBL23-C1 from www.mouser.com. How much energy is stored in a cubic meter of
such batteries? At Mouser’s prices, what does a cubic meter of batteries cost? For
quantity cost you can use the price of a very similar battery the UBBL23-F1 for which
Mouser gives a volume discount.
Make a table of these numbers in the SI units of Joules per cubic meter and include the
number from magnetic storage as well.
I think you might find http://www.convertunits.com/ a useful resource.
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