Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

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1 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field is ceated by mass. In Chapte 19 we saw something simila: electic field is ceated by electic chage. Fo these two situations we acknowledged that mass and chage wee both fundamental popeties of matte... i.e. all matte has mass and all matte has chage. We used the simplest bit of mass o chage, a small paticle with spheical symmety, to wite ou basic expession fo the gavitational field due to a bit of mass, o the electic field due to a bit of chage: Gm k q e g = E =. whee is the distance fom the mass o chage to the point at which we ae evaluating the stength of the gavitational field o electic field. The facto of aises due to the fact that the stength of the field speads out in the shape of a sphee, and the aea of a sphee is popotional to. So the stength of the field diminishes in a way that is invesely popotional to. In 180, Hans Oested accidentally discoveed that electic cuent ceates magnetic field. So we can add magnetic field to ou list of fields in natue that ae esponsible fo applying foces: Mass ceates gavitational field. Electic chage ceates electic field. Electic cuent ceates magnetic field. Thee ae, howeve, some notable diffeences between the fist two of these and the last. While mass and chage ae fundamental popeties of matte, electic cuent is not. The simplest bit of mass o chage is a small sphee, as illustated above; but we cannot have a small, isolated bit of cuent in the Page 1 of 6

2 Physics B Chapte Notes - Magnetic Field Sping 018 same way. Fo these easons, the expession fo the magnetic field due to a cuent is simila to the expessions above, but slightly diffeent to allow fo these diffeences. We define the simplest kind of cuent as that of a long, staight, cuent-caying wie. That is, a wie that extends vey fa in both diections... fa enough that we cannot easonably see eithe end of the wie. Fo such a wie, we can define the stength of the magnetic field at a point a distance fom the wie: B = k m I magnetic field a distance fom a long, staight, cuent-caying wie The most significant diffeence between this expession and those fo the gavitational o electic field is the lack of a squae on the in the denominato. This is due to geomety: while g and E spead out in the shape of a sphee, the magnetic field speads out as a cicle; it does not spead out in the dimension paallel to the cuent. As such, the aea ove which it speads out (i.e. essentially the aea of a cylinde and not of a sphee...) is popotional to the cicumfeence of the cicle, and so it is popotional to the adius of the cicle (and not to.) As the standad unit of magnetic field is the Tesla, abbeviated with a T, the value of the constant in this expession is: k m = x 10-7 T-m/A = 0. µt-m/a. Diection of the Magnetic Field Anothe significant way in which the popeties of the magnetic field diffe fom the popeties of the gavitational o electic fields is how we detemine the diection of the magnetic field. The diection of the gavitational o electic field in the illustation above is simple: the gavitational field points diectly towad Page of 6

3 Physics B Chapte Notes - Magnetic Field Sping 018 the mass that ceates it, and the electic field points diectly away fom the chage that ceates it (assuming the chage is positive.) The diection of the magnetic field, howeve, is neithe towad no away fom the cuent that ceates it. Unlike mass o chage, the cuent itself has diection. And the diection of the coesponding magnetic field is always pependicula to the diection of the cuent. We also find that the diection of the magnetic field is pependicula to, which is fom the cuent to the point at which we want to evaluate the magnetic field. We can illustate the possible diections of the magnetic field fo a cuent of a long, staight, cuent-caying wie by imagining the cuent pointing towad us (i.e. out of the page) and consideing seveal to diffeent locations: We can define the diection of B elative to the diections of the cuent and by using the ight hand ule fo a vecto poduct (also known as a coss poduct.) To use the ight hand ule: Stat with the finges of you ight hand flat and you thumb out, in the shape of an L. With you finges flat, place them in the diection of the cuent. Keeping you thumb out, fold you finges in the diection of. You thumb points in the diection of the magnetic field. To apply the ight hand ule to the diagam above, you finges would point out of the page (i.e. the diection of the cuent.) As you otate you hand to allow youself to fold you finges towad each, you should find that you thumb points in the diection of B fo each coesponding. Thee is a simplified vesion of this pocess that also involves using you ight hand: Cul you finges, stick you thumb out. Place you thumb in the diection of the cuent. Rotate you culed finges until they ae in the position designated by. You finge tips should now point in the diection of the magnetic field. (Ty this fo the positions in the illustation above.) Page 3 of 6

4 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Segment of Wie We used the idea of a long, staight wie above because it is simple and the coesponding expession fo the magnetic field is also simple. Howeve, not all wies ae long o staight, so we now must conside the magnetic field due to a wie that is not long. Conside the wie in the diagam below: The boken lines indicate the wie (and the cuent) extend vey fa in those diections. We would like to conside the magnetic field ceated by this cuent at point P. We fist conside segment 1. Note that fo the cuent in this pat of the wie, the diection of the cuent is the same as the diection of (i.e. in this case, the diection of both is vetical.) This means it is impossible to use the ight hand ule to detemine the diection of the magnetic field at point P. Thee is a simple conclusion: If the cuent and ae in the same diection, as they ae fo segment 1 above, the magnetic field fom that cuent at that point is zeo! So in this situation, the magnetic field fom segment 1 is zeo. What about segment? Notice that segment begins at a point that is diectly unde point P and extends vey fa in only one diection. Essentially, segment is half of a long, staight, cuent-caying wie. Due to symmety, we can claim that the contibution to the magnetic field fom segment is exactly half that of a l.s.c.c.w. ; the othe half would have come fom the left half of the l.s.c.c.w., which is not pesent in this illustation. Fo this eason, we can claim that fo a semi-infinite wie, i.e. a wie that stats at a point diectly adjacent to the point at which we want to evaluate the magnetic field and extends vey fa in only one diection, the magnetic field must be one half of the expession fo a l.s.c.c.w. : B = 1 k m I magnetic field a distance fom a semi-infinite wie Page 4 of 6

5 Physics B Chapte Notes - Magnetic Field Sping 018 Now we can go one step futhe and conside the magnetic field due to a finite staight segment of wie. Conside the thee segments of the wie in the illustation hee: We aleady know that the contibution to the magnetic field at point P fom segment 1 is zeo. What about segment? We can define an angle θ by dawing a line fom point P to the fa end of segment. Note that if segment was longe, the angle θ would be close to 90 O. If segment was so long as to be consideed a semi-infinite wie, the angle θ would essentially be 90 O. And if segment was vey shot, it would essentially cease to exist and the angle θ would be zeo. Note that ou segment in this illustation is a faction of the semi-infinite wie pesented peviously. What faction? The angle θ povides a clue: when the angle is zeo, the faction is zeo and when the angle is 90 O, the faction is one. This behavio descibes a sine function, and in fact the magnetic field due to the cuent in this segment is a faction of the magnetic field fom a semi-infinite wie. The faction is defined by sinθ, and so: B = 1 k m I sinθ magnetic field a distance fom a segment of cuent Note: I wote this expession using simple logic, but of couse that s not the pope way to deive an expession. The pope way in this situation would be to use the Biot-Savat law, which would equie a simple integal, and would etun the same esult. Magnetic Field fom a Cicula Loop of Wie We used the idea of a long, staight wie above because it is simple and the coesponding expession fo the magnetic field is also simple. Howeve, not all wies ae long o staight, so we now can conside the magnetic field at the cente of a cicula cuent loop. The deivation of the expession fo this Page 5 of 6

6 Physics B Chapte Notes - Magnetic Field Sping 018 equies use of the Biot-Savat law, which consides the contibution to the magnetic field of evey tiny segment of cuent. In the case of a cicula loop of cuent, we can see that (using the ight hand ule fo I coss, i.e. cuent coss...) the diection of the magnetic field at the cente due to evey bit of cuent points in the same diection: into the page if the cuent is clockwise, out of the page if the cuent is counteclockwise. I will skip the details hee, but using the Biot-Savat law to find the magnitude of the magnetic field at the cente of a cuent loop we get: kmπ I B = magnetic field at cente of a cuent loop of adius The diection of the magnetic field is expessed above: into the page if the cuent is clockwise, out of the page if the cuent is counteclockwise. A quick and simple way to detemine the diection of the magnetic field is available once again by using the ight hand: cul you finges, stick you thumb out; oient you finges in the diection of the cuent aound the loop, and you thumb will point in the diection of the magnetic field at the cente of the loop. (Ty this fo the two loops in the illustation above.) If a wie includes a segment that is a faction of a cicula loop, the magnetic field at the cente of cuvatue of that segment is simply the expession given hee multiplied by the elevant faction (i.e. whateve faction of a full loop is epesented by the segment.) Page 6 of 6

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