this video we qualitatively discuss the property called self-inductance for a solenoid
then we derive and apply the formula for the self-inductance to an example as a quick reminder
in the picture on the left we have a solenoid that's just a long coil of wire and we previously
discussed that a current carrying solenoid has a uniform magnetic field inside and that's given
by the formula b equals mu naught n i where mu naught is called the permeability of free space
little n is the number of turns per unit length or turns per meter for the coil and i of course
is the current but now that we know a little bit about induction we realize that connecting a
solenoid to a power source creates a bunch of new flux through the turns in the coil that's going to
induce an emf fighting the change in flux through each of those turns and this is the idea of lenses
law this induced emf will then slow the increase in current as the solenoid gets up and running
conversely once current is running in a solenoid if we suddenly switch off the power source an emf
will be induced that tries to maintain the current flowing in the same direction it was to begin with
because it's fighting any change of flux through the coils this resistance to changes in current
is quantified as the self-inductance of the coil so how do we measure self-inductance
we're going to start from faraday's law and faraday's law is quantifying the induced emf
if i have a known change in flux for each of the loops in the coil n is the number
of loops in the coil and delta t of course is the time for the change in flux to occur
but this delta phi in the coil that's again the change in flux per loop so for an individual loop
and we multiply by n later in faraday's law the change in flux is certainly related to the current
in the coil and i'm going to write it first as a times b final minus a times b
initial there i'm using the formula for flux that is given by the area of the loop
times the intensity of the magnetic field and when the magnetic field is perpendicular to
the loops like it is in this case it's that simple to measure flux but then i can relate
the field inside this coil to the current running through it by using our first formula
and i find that my change in flux is given by a mu naught n times the change in the current
this allows me to go back to faraday's law and change the appearance of that a little
bit and our goal here by the way is to find a relationship between the induced voltage and the
rate of change in the current so i replace delta phi with a mu naught n times delta i
and now i've got an induced emf in the coil as being proportional to the rate of change in the
current there's one more modification to make to get this in its final form instead of writing
little n in here i can re-express it as the total number of turns divided by the length of the coil
and that allows me to combine two factors of capital n the total number of turns
so i get negative mu naught n squared a
over the length of the coil and i'm going to use a lowercase l for this
because we need capital l for something else
times delta i over delta t i'll just change that l over real quick
so this constant of proportionality when i get the induced emf proportional
to the rate of change in current the constant of proportionality is called
the inductance or self-inductance of the coil and that's a way of quantifying how intensely
this coil is going to resist changes in current if my self-inductance is bigger
it means for a given change in current i'm inducing a bigger emf to resist that change
and we commonly use the symbol capital l for self inductance so for a solenoid capital l is given by
mu naught n squared a over l little l for the length of the solenoid
so the units of inductance can be expressed in terms of this induced emf that has units
of volts and over here on the right hand side my change in current over change in time that's amps
per second so for the self-inductance to produce volts out of this i need volt seconds per amp
as the units and there's a special name for this it's called the henry and it's a capital h finally
i can rework my induced emf formula and if i just talk about magnitudes here and ignore the minus
sign the minus sign is really just telling us that the induced emf always resists the change in flux
if i just write down a formula in terms of magnitude it's just this
the induced emf is given by the inductance l times the rate of change in the current and again a
bigger l means you get a more intense induced emf resisting the change in current and i can also see
from the formula the big rate of change in the current induces the larger emf which makes a lot
of sense so let's wrap things up by applying our inductance formula to a solenoid and we're given
a solenoid with a length of 10 centimeters a number of turns 2 900 average diameter
of the coils 2.9 centimeters and of course we can use that to find the area of those coils
area will be one quarter pi diameter squared that's the same thing as writing down pi r squared
and i get one quarter pi times 0.029 squared and that comes out to 6.61 times 10 to the
negative 4 square meters then i can compute the self-inductance of this coil using our new formula
that's mu naught n squared a over l
and i'll crunch those numbers and i get a self-inductance of .0699 henrys and you
would normally express that in millihenries as 69.9 millihenries next we're told that this coil
had a dc current of half an amp flowing through it and we're trying to compute the induced emf
if we shut the current down in 100th of a second by unplugging the solenoid so in this case we're
talking about the emf that's induced to attempt to keep the current flowing in the coil and we use
our new formula for the induced emf in terms of the self-inductance that's l delta i over delta t
my self-inductance is .0699 henry's my change in current is half an amp
and my change in time we shut this thing off in 100th of a second and when i crunch the
numbers i get 3.50 volts induced to try to keep this current flowing this by the way
is the reason why when you unplug appliances that are already running you frequently will get
a spark jumping from the outlet to the cord it's that current attempting to continue moving even
as you're shutting things down if you find the physics content on zack's lab helpful click on the
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