PH -C Fall 1 Magnetic Fields Due to Cuents Lectue 14 Chapte 9 (Halliday/esnick/Walke, Fundamentals of Physics 8 th edition) 1
Chapte 9 Magnetic Fields Due to Cuents In this chapte we will exploe the elationship between an electic cuent and the magnetic field it geneates in the space aound it. We will follow a two-ponged appoach, depending on the symmety of the poblem. Fo poblems with low symmety we will use the law of Biot-Savat in combination with the pinciple of supeposition. Fo poblems with high symmety we will intoduce Ampee s law. Both appoaches will be used to exploe the magnetic field geneated by cuents in a vaiety of geometies (staight wie, wie loop, solenoid coil, tooid coil).we will also detemine the foce between two paallel, cuent-caying conductos. We will then use this foce to define the SI unit fo electic cuent (the ampee).
i ds db 3 4 A The Law of Biot - Savat This law gives the magnetic field db geneated by a wie segment of length ds that caies a cuent i. Conside the geomety shown in the figue. Associated with the element ds we define an associated vecto ds that has magnitude equal to the length ds. The diection of ds is the same as that of the cuent that flows though segment ds. The magnetic field db geneated at point P by the element ds located at point A i ds is given by the equation db. Hee is the vecto that connects 3 4 point A (location of element ds) with point P at which we want to detemine db. pemeability constant. db db 7 6 The constant 4 1 T m/a 1.6 1 T m/a and is known as the i ds sin 4 " " The magnitude of is. Hee is the angle between ds and. 3
Magnetic Field Geneated by a Long Staight Wie The magnitude of the magnetic field geneated by the wie at point P located at a distance fom the wie B i is given by the equation i B. The magnetic field lines fom cicles that have thei centes at the wie. The magnetic field vecto B is tangent to the magnetic field lines. The sense fo B is given by the ight - hand ule. We point the thumb of the ight hand in the diection of the cuent. The diection along which the finges of the ight hand cul aound the wie gives the diection of B. 4
i B B x a 3/ Poof of the equation. Conside the wie element of length ds shown in the figue. The element geneates at point P a magnetic field of i ds sin magnitude db. Vecto db is pointing 4 into the page. The magnetic field geneated by the whole wie is found by integation: i B db db ds sin s sin sin / / s i ds i s 3/ s s i dx x a x a 5
B. db. Magnetic Field Geneated by a Cicula Wie Ac of adius at Its Cente C A wie section of length ds geneates at the cente C a magnetic field db. i ds sin 9 i ds The magnitude db. The length ds d 4 4 i db d. Vecto db points out of the page. 4 The i net magnetic field BdB d = 4 Note : The angle must be expessed in adians. Fo a cicula wie,. In this case we get: i. 4 B B cic i 4 i. 6
Magnetic Field Geneated by a Cicula Loop Along the Loop Axis The wie element ds geneates a magnetic field db i ds sin 9 i ds whose magnitude db. 4 4 We decompose db into two components: One ( db ) is along the z-axis.the second component ( db ) is in a diection pependicula to the z-axis. The sum of all the db is equal to zeo. Thus we sum only the db tems: i ds cos z db db 4 i ds i ds db B db 3 3/ 4 4 z cos cos z z i i i B ds 4 4 z 3/ 3/ 3/ 7
Example F 1 I L B1 µo- pemiability constant 8
Supeposition Pinciple 9
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B ds i enc Ampee's Law The law of Biot-Savat combined with the pinciple of supeposition can be used to detemine B if we know the distibution of cuents. In situations that have high symmety we can use Ampee's law instead, because it is simple to apply. Ampee's law can be deived fom the law of Biot-Savat, with which it is mathematically equivalent. Ampee's law is moe suitable fo advanced fomulations of electomagnetism. It can be expessed as follows: The line integal B ds of the magnetic field B along any closed path is equal to the total cuent enclosed inside the path multiplied by. The closed path used is known as an " Ampeian loop. " In its pesent fom Ampee's law is not complete. A missing tem was added by Clak Maxwell. The complete fom of Ampee's law will be discussed in Chapte 3. 11
B ds i enc Implementation of Ampee's Law : 1. Detemination of B ds. The closed path is divided into n elements s, s,..., s. 1 We then fom the sum: n n B s Bs cos. Hee B is the i i i i i i i1 i1 magnetic field in the ith element. n Bds lim B s as n i1. Calculation of i i enc i. We cul the finges of the ight hand in the diection in which the Ampeian loop was tavesed. We note the diection of the thumb. All cuents inside the loop paallel to the thumb ae counted as positive. All cuents inside the loop antipaallel to the thumb ae counted as negative. All cuents outside the loop ae not counted. In this example : i i i. enc 1 n 1
Magnetic Field Outside a Long Staight Wie We aleady have seen that the magnetic field lines of the magnetic field geneated by a long staight wie that caies a cuent i have the fom of cicles, which ae concentic with the wie. We choose an Ampeian loop that eflects the cylindical symmety of the poblem. The loop is also a cicle of adius that has its cente on the wie. The magnetic field is tangent to the loop and has a constant magnitude B : Bds Bdscos B ds B i i enc i B Note : Ampee's law holds tue fo any closed path. We choose to use the path that makes thecalculation of B as easy as possible. 13
i B Magnetic Field Inside a Long Staight Wie We assume that the distibution of the cuent within the coss-section of the wie is unifom. The wie caies a cuent i and has adius. We choose an Ampeian loop that is a cicle of adius ( ) with its cente on the wie. The magnetic field is tangent to the loop and has a constant magnitude B : Bds Bdscos B ds B i ienc i i B i i B enc O 14
The Solenoid The solenoid is a long, tightly wound helical wie coil in which the coil length is much lage than the coil diamete. Viewing the solenoid as a collection of single cicula loops, one can see that the magnetic field inside is appoximately unifom. The magnetic field inside the solenoid is paallel to the solenoid axis. The sense of B can be detemined using the ight-hand ule. We cul the finges of the ight hand along the diection of the cuent in the coil windings. The thumb of the ight hand points along B. The magnetic field outside the solenoid is much weake and can be 15 taken to be appoximately zeo.
B ni We will use Ampee's law to detemine the magnetic field inside a solenoid. We assume that the magnetic field is unifom inside the solenoid and zeo outside. We assume that the solenoid has n tuns pe unit length. We will use the Ampeian loop abcd. It is a ectangle with its long side paallel to the solenoid axis. One long side ( ab) is inside the solenoid, while the othe ( cd) is outside: b Bds Bds a B ds Bds cos B ds Bh B ds B ds B ds B ds Bh The enclosed cuent ienc nhi. B ds i Bh nhi B ni ec n Bds Bds Bds c d a b c d b b b c d a a a a b c d 16
B Ni o Magnetic Field of a Tooid A tooid has the shape of a doughnut (see figue). We assume that the tooid caies a cuent i and that it has N windings. The magnetic field lines inside the tooid fom cicles that ae concentic with the tooid cente. The magnetic field vecto is tangent to these lines. The sense of B can be found using the ight-hand ule. We cul the finges of the ight hand along the diection of the cuent in the coil windings. The thumb of the ight hand points along B. The magnetic field outside the solenoid is appoximately zeo. We use an Ampeian loop that is a cicle of adius (oange cicle in the figue): Bds Bdscos B ds B. The enclosed cuent i Ni. Ni Thus: B Ni B. Note : The magnetic field inside a tooid is not unifom. enc 17
The Magnetic Field of a Magnetic Dipole. Conside the magnetic field geneated by a wie coil of adius that caies a cuent i. The magnetic field at a point P on the z-axis is given by Bz ( ) 3 z B i z 3/. Hee z is the distance between P and the coil cente. Fo points fa fom the loop i ( z ) we can use the appoximation: B 3 z i ia B. Hee is the magnetic 3 3 3 z z z dipole moment of the loop. In vecto fom: Bz ( ). 3 z The loop geneates a magnetic field that has the same fom as the field geneated by a ba magnet. 18