in this video we discuss the magnetic field generated by a current carrying wire
and the field generated inside of a solenoid or a coil of wire we begin with the current carrying
wire and the magnetic field generated by this wire is really interesting it swirls around the wire
according to a right hand rule you put the thumb of your right hand in the direction of the current
and your fingers on your right hand will curl in the direction of the field the strength of the
magnetic field around this wire of course drops off as you get farther away from it
and this is quantified by ampere's law and that's mu naught i over 2 pi r where mu naught is called
the permeability of free space and it's given by 4 pi times 10 to the negative 7 tesla meters per amp
you can verify the units make sense real quick by plugging into the formula mu naught has units of
teslas in it but i have to cancel one unit of amps because of this i here for the current and i have
to cancel the meters in the denominator from this r which is the separation distance from the wire
so the meters and amps cancel out and i get teslas note that the definition of mu naught
changed very slightly in 2018 which changed it about nine decimal places in but we're going
to stick with this traditional 4 pi times 10 to the negative 7 because it's easier to remember
in our example we want the magnetic field strength and direction at a distance of 50
centimeters above and below the wire shown below so i put a 2.5 amp current through this wire and i
need the magnetic field direction above and below the wire and that's done with the right hand rule
i put my thumb in the direction of the current the field wraps around in the direction of my fingers
and so directly above this wire my fingers are poking out of the page
so my magnetic field is outward and then below the wire the magnetic field is diving back
in to the page and we use x's for that as for the magnitude we use ampere's law for this and we just
plug in the numbers four pi times ten to the negative seven a current of two and a half amps
divided by two pi times the distance from the wire which is half a meter
and we get 10 to the negative six teslas for this field on the right side of the screen we're
talking about the field generated by a solenoid so this is a perfect cylindrical coil of wire
i've shown the current running down the face that's closest to us which means it's running
up the face that's farthest from us and coming out of the page at the top of the solenoid and
into the page at the bottom of the solenoid so to get a better sense for what's going on
in the second picture i slice the whole thing in half and i can see that current going into the
page all along the bottom of the solenoid every time a wire dives through the plane of the page
and the current's coming out at the top of the solenoid now what i've done in the picture is i've
applied the right hand rule at all those points where the current's going in or going out so if i
apply the right hand rule to these inward currents i put my thumb into the page and my fingers curl
in the direction of the field around each of those wires and that's what all these little loops are
same for the top of the solenoid i put my thumb out of the page
and my fingers curl in the direction of the field so that's what all these little loops are
now we're not interested in the field outside of the solenoid actually if the solenoid is very long
and skinny the field outside is very nearly zero what we're interested in is the field generated
inside the solenoid and i can see from all these little loops around the wires all these little
contributions along the upper edge those are all pointed rightward with a little bit of extra tilt
and i can see that where one of the loops has a little upward component like right here
the other one next to it will have a little downward component and those actually cancel
out perfectly and it leads to an exactly rightward pointing magnetic field inside the solenoid
and in fact you can prove that it's a constant field so even in the middle of the solenoid it
has exactly the same strength as it does toward the edges where the wires are so there's the
constant magnetic field inside the solenoid at the ends of the solenoid you will see fringing
and that fringing looks very much like the field of a permanent magnet made of iron or some other
ferromagnetic material and in fact that's why a solenoid is called an electromagnet this is just a
way of electrically generating with a coil of wire a field that looks just like a permanent magnet
and this has a lot of advantages because you can just flip on a switch and turn it into a strong
magnet but when you don't need it to be a magnet you just turn off the switch and it's not a magnet
anymore now the strength of the field inside the solenoid is given by this little formula b equals
mu naught n i where mu naught is the permeability of free space that we defined back here
i is of course the current through the wire and the little n deserves some explanation
it's actually the number of coils per unit length in other words coils per meter in the solenoid
i could also call it loops per meter so if you double the density of the loops in your solenoid
maybe wrap two layers of loops on top of each other then of course that would double the field
inside the solenoid let's do a quick example i have a solenoid formed by wrapping a thousand
turns of wire on a five centimeter length that allows us to calculate the little n so
i have a thousand turns or loops on a length of point zero five meters i simplify that to
get turns per meter and that's 20 000 turns per meter then i want the field strength inside the
solenoid if we run 0.25 amps of current through it so i just plug into b equals mu naught and i
and that's four pi times ten to the negative seven twenty thousand for the turn density
and then zero 0.25 amps for the current and i get .0063 just keeping two sig figs teslas or
i could call this 6.3 miller tesla if you find the physics content on zak's lab helpful click on the
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