Calculating Magnetic Fields Around Straight Current-Carrying Wires

a current of 15 amps flows north along a wire so let's draw a picture

so let's say if we have a wire or even a conductor and the current is flowing in this

direction calculate the magnitude and direction of the magnetic field

at a point 10 centimeters east of the wire so east is this direction

so 10 centimeters away from it we wish to calculate the magnetic field at that point

now whenever you have a straight wire to determine the direction of the magnetic field around that wire here's

what you can do what you need to do is take a pen

and take your right hand and curl your fingers around the wire and make sure that the current

is in the direction as your thumb let's say if we have a current going in this direction

and if you curl your fingers around that wire with your thumb pointed to the right

your fingers should flow in this general

direction so above the pen notice that the magnetic field is coming

out of the page represented by that symbol below the pen or below the wire it's

going into the page now let's use that same technique for this wire

so take your right hand curl your fingers around the pen and make sure your thumb is pointing up

in the direction of the current so on the left side the magnetic field should be coming

out of the page and then on the right side it should be going into the page

if you do it correctly so we're going to have an x on the right

and the dot on the left this is out of the page and x represents in the page

so at some point a 10 centimeters to the right of the wire we now know the direction of the

magnetic field is going into the page so all we need to do is calculate the magnitude

of the magnetic field now those of you who are having trouble with that right hand rule

you can see it this way so you want to point your thumb in the direction of the current

and then make sure your fingers curl around the wire

my drawing is not perfect but hopefully you can see at this point so here's the direction of the current

and notice that your fingers behind the wire curl in this direction

and so it's out of the page on the left into the page on the right so hopefully you can see that visual illustration

now let's calculate the magnitude of the magnetic field and the formula that we need is this equation

it's equal to mu zero times the current divided by two pi r and so the formula is pretty simple

and so this is equal to four pi times ten to negative seven and i represents the current it's 15

amps and r is basically the distance between the wire and a point of interest

so r is 10 centimeters which is 0.1 meters so let's go ahead and get the answer

so you should get 3 times 10 to the negative 5 tesla

and so that's it so as you move away from the wire the strength of the magnetic field

greatly decreases as r increases the magnetic field decreases

now if you wish to increase the strength of the magnetic field you can move closer to the wire or you

can increase the current increase in the current increases the strength of the magnetic field

proportionally so if you double the current the magnetic field will double in value if

you triple the current the magnetic field will triple if you double the distance the magnetic

field will be reduced to one half of its original value if you triple the distance it's going to be one third of

its value if you decrease the distance by a factor of two if you basically cut in half the

magnetic field will increase by a factor of two so this is the format to calculate

the magnetic field due to a long straight wire now let's move on to the next problem

a current of 20 amps flows east along a wider another current of 30 amps flows west

five centimeters beneath the first wire calculate the magnetic field at a position of two centimeters above and

below the first wire so let's say that this is the first wire and this is going to be

the second wire so in the first wire we have a current of

20 amps flowing east and let's call this i1

now in the second wire we have a current of 30 amps which is flowing west

and so let's call this i2 now what we need to do is calculate the

magnetic field two centimeters above the first wire so let's calculate it here and let's call that point a

so that's out of position two centimeters above the first wire and the second wire is five centimeters

beneath the first wire so feel free to pause the video and

calculate the magnetic field at that position now first we need to determine the

direction of the magnetic field so if you use the right hand rule and for the first wire if you curl your

fingers around the wire with your thumb pointing to the right the magnetic field should be

going in this direction like this so what that means is that

above the wire it's coming out of the page and below the wire it's going into the

page and we're going to call this b1 that's the magnetic field created due to

current one now wire two the current that flows in it

also creates a magnetic field anytime a charge is moving it creates its own magnetic field

now because the other wire is going in the opposite direction if you curl your hands

around the wire and with your thumb face to the left this time the current should be going any i mean

the magnetic field rather should be going in the opposite direction let me use a different color

so it should look something like this and so notice that it's going into the

page at the top and it's coming out of the page at the bottom so this is b2

so notice that apposition a at this point b1

is going into the page but b2 anywhere above wire two including this

position b2 is going uh into the page b1 is coming out of the page so this is out of the page and this is into the page in

case if i somehow mix those terms so now that we know b1 is going out of

the page and b2 is instapage we can now calculate the net magnetic field at point a

so notice that these two have opposite signs so if it's coming out of the page that

means b1 is positive if it's going into the page like b2 that's negative that's in a negative z

direction b1 is in a positive z direction

so at point a then that magnetic field is the sum of b1 and b2 so b1 is positive because it's coming

out of the page and b2 is negative because

it's going into the page so this is going to be b1 minus b2 b1 is equal to mu zero that's the

permeability of free space times i1 divided by 2 pi r1

b2 is going to be mu 0 i2 over 2 pi r2

so now we could factor out mu zero and also two pi and so we're going to be left with i1

over r1 minus i2 over r2

and so this is going to give us the net magnetic field generated by both wires so mu zero is four pi

times ten to the minus seven and then we can divide that by two pi the first current

is 20 amps and r1 the distance between current one and point a that's two

centimeters but if we divide that by 100 that's going to be 0.02 meters

i2 is 30 amps and r2 that's the distance between point a

and current two so that's going to be five centimeters plus two centimeters or

seven centimeters which is point zero seven meters

now four pi divided by two pi is simply two so this becomes two times ten to the minus seven

and then we have twenty divided by point zero two so that's a thousand

and minus thirty divided by point zero seven which is

four hundred twenty eight point five seven one so now

let's subtract a thousand by that number and then multiply that result by two times ten to the minus seven

so as your answer you should get this the net magnetic field is

1.14 times 10 to the negative 4 tesla

so that's the net magnetic field of both wires at point a now let's move on to the second part

so we need to calculate the net magnetic field two centimeters below the first wire

so that's going to be somewhere over here let's call this

point b now notice that b1 and b2 they're both going into the

page at point b so they're both negative which means the

net magnetic field has to be negative so i'm going to put a negative sign on

both of them so we're adding two negative terms negative b1

plus negative b2 or simply negative b1 minus b2 so that's going to be negative

mu zero i1 over 2 pi r1 minus mu 0 i2

over 2 pi r2 so just like before we're going to take out negative mu 0 over 2 pi

and so we'll be left with i1 over r1 minus i2

divided by r2 so mu 0 that's negative 4 pi times 10 to the minus 7 divided by 2 pi

and then i1 is 20 now r1 the distance between

the first wire and point b it's two centimeters below it so r1 is going to be two again

or 0.02 meters now the only thing that's different really is r2

because point b is three centimeters above the second wire it's five minus two and

that's how we get three now there's one mistake i need to correct

i factored out negative mu zero over two pi so this is now positive which makes this

positive so let's just make that correction

negative four pi divided by two pi is negative two so this is negative two times ten to the

minus seven and then twenty divided by point zero two

that's a thousand and then thirty divided by point zero three that's a

thousand as well so a thousand plus a thousand is two thousand and if you take two thousand

multiply by negative two times ten to negative seven that's equal to negative four

times 10 to the negative 4 tesla so that's the net magnetic field the negative sign tells us that the net

magnetic field is going in the negative z direction or into the page and that's it

so now you know how to calculate the magnetic field due to a long straight wire or if you have two long straight

wires so you just got to determine the direction of the magnetic field and if

you need to add or subtract them and just remember if it's going out of the page

it's positive it's going in the positive z direction if it's going into the page it's negative since it's going in the

negative z direction you