Chapter F. Magnetism. Blinn College  Physics Terry Honan


 Janis Carr
 1 years ago
 Views:
Transcription
1 Chapte F Magnetism Blinn College  Physics 46  Tey Honan F.  Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S. Playing with ba magnets demonstates that like poles epel and unlike attact. This is analogous to the situation we had with electic chage. This analogy is a deep one and is called Electomagnetic duality. Noth and South poles ae elated to the magnetic field B as positive and negative electic chages ae to the electic field. N and S ae to B as + and  ae to E The foce of an electic chage q in an electic field gives by analogy the foce of a magnetic pole of stength g in a magnetic field. F = q E and F = g B If g denotes the "magnetic chage" o pole stength then a Noth pole coesponds positive g and a South pole to negative. Gauss's law elates the total electic flux though a closed suface to the chage enclosed by the suface. Similaly, the magnetic flux though a closed suface coesponds to the total "magnetic chage" inside. E ÿ A = ε 0 q enclosed and B ÿ A = m 0 g enclosed whee analogous to ê ε 0, the electic constant of popotionality, is the magnetic constant m 0. E = q ` 4 p ε 0 and B = m 0 4 p g ` Magnetic Dipoles A pemanent magnet has both Noth and South poles sepaated by some distance. When it is placed in a field the N pole expeiences a foce in the diection of the field and the S pole has a foce opposite the field. If the field is unifom the net foce is zeo but thee is a net toque. This is analogous to an electic dipole and it will be called a magnetic dipole. The stength of a magnet can be descibed by its magnetic dipole moment m. Fo an electic dipole p the toque and potential enegy given by t = p äe and U = p E. The coesponding expessions fo toque and potential enegy of a magnetic dipole in a magnetic field ae t = m äb and U = m B. In addition to pemanent magnets being dipoles we will see that cuent loops also ae magnetic dipoles. In that case we will wite an expession fo the magnetic dipole moment. Gauss's Law fo Magnetism and the Absence of Isolated Poles Isolated magnetic poles could exist but so fa none have eve been obseved. Gauss's law in the electic case states that electic field lines begin at isolated positive chages and end at isolated negative chages. The absence of isolated magnetic poles implies that magnetic field lines neve begin o end; they eithe fom closed loops o go off to infinity The magnetic analog of Gauss's law is ò B ÿ A = m 0 g inside. The nonexistence of isolated magnetic poles implies that the ight hand side is zeo and Gauss's law fo magnetism becomes B ÿ A = 0 This is ou second of Maxwell's equations. If we apply Gauss's law to a magnet and put a Gaussian suface aound the Noth pole then thee is magnetic flux leaving the suface at the end of the magnet. Fo the flux to be zeo though this Gaussian suface the field lines inside the magnet
2 Chapte F  Magnetism magnetic flux leaving the suface at the end of the magnet. Fo the flux to be zeo though this Gaussian suface the field lines inside the magnet must close back on themselves and fom closed loops. Because of this if a ba magnet is cut in half then it doesn't split into a pai of isolated poles; it becomes two smalle dipoles. If someday isolated poles ae discoveed, then we may just modify Gauss's law by adding in a magnetic chage tem m 0 g inside to its ight hand side as was shown above. Othe modifications to Maxwell's equations associated with magnetic cuents will also be needed; these will be discussed late. F.  Foce on Moving (Electic) Chages and Cuents Electicity and magnetism ae not sepaate foces whee electic fields just exet foces on electic chages and magnetic fields exet foces on magnets. Instead electicity and magnetism ae aspects of the same foce called electomagnetism. Magnetic fields cause foces on moving (electic) chages and cuents. Magnetic Foce on Moving Chages If a chage Q is moving with a velocity v in a magnetic field B then the foce is given by F = Q v äb. Note that the coss poduct is a thee dimensional thing. The velocity and field vectos define a plane and the foce is in the diection pependicula to the plane. Note also that when a vecto is multiplied by a negative scala its diection changes, so negative chages expeience foces opposite that of positive ones. Foce on Cuents Conside the flow of chage caies of chage q with dift velocity v d though a staight segment of wie of length { with cosssection A. If the density of chage caies (numbe/volume) is n then the total numbe of chage caies is N = n µvolume = n A {. Summing ove all the chages gives the total foce on the wie in a unifom field F = N q v d äb = n A { q v d äb Using the expession fo cuent fom the pevious chapte, I = q n A v d, we get F = I { äb, whee the diection of the cuent is put into the diection of the vecto {. Note that the cuent diection is the same as the dift velocity when q is positive and it is opposite when q is negative. This is built into the above expession. To genealize this expession conside a cuved wie with an infinitesimal segment s. The foce on that segment is I s äb. Integating ove the length of the wie gives whee the field need not be unifom. F = I s äb, Hall Effect It is clea fom the pevious section that we cannot detemine the chage of the chage caies by measuing the magnetic foce on a wie; simultaneously changing the signs of q and v d gives the same cuent. We can, howeve, use the magnetic foce to find the chage of the chage caies by measuing the voltage acoss a conducting stip in a magnetic field. Conside a flat conducting stip with a cuent in a magnetic field. Take the width of the stip, the cuent and the field to be mutually pependicula as shown in the diagam.
3 Chapte F  Magnetism 3 B F I F v q < 0 q > 0 d v d L The magnetic foce will push eithe positive o negative chage caies towad the top of the wie. This will ceate a voltage acoss the stip. The polaity of the voltage depends on the chage of the chage of the chage caies. If they ae positive then the top is positive and othewise it is negative. The Hall voltage is the voltage acoss the stip. If the width of the stip is L then the wok on the chage caie q is F L. Since the foce is F = q v B we have The wok pe chage is the induced voltage, since DU = qdv. W = q v d B L V Hall = W êq ï V Hall = v d B L Motion of Chaged Paticles Any foce that acts pependiculaly to the velocity of a paticle doesn't affect the speed of the paticle; it only altes its diection. This is the case with the magnetic foce F = Q v äb. Suppose a paticle with speed v is shot into a egion of unifom magnetic field with the velocity pependicula to the field then the magnitude of the foce is just F = Q v B. Since the speed and the magnitude of the foce ae constant and the foce and velocity ae pependicula, the motion will be unifom cicula motion. Using the acceleation fo unifom cicula motion a c = v ê and Newton's second law we get: F = m a ï Q v B = m v ï = m v Q B. The angula fequency w is elated to the speed and adius by w = vê which gives an expession known as the cycloton fequency w = Q B m. If a chaged paticle moves in a unifom magnetic field with a velocity that is not pependicula to the field, then the pependicula component changes as befoe and the paallel component is unchanged. The esulting motion is a combination of linea and cicula motion, giving a helix. The geneal shape of the path of a chaged paticle in a unifom magnetic field is helical. An electomagnetic field is a combination of both electic and magnetic fields. The foce of a chaged paticle in an electomagnetic field is the sum of both electic and magnetic foces and is called the Loentz foce law F = Q IE + v äbm. F.3  Souces of Magnetic Fields In ou discussion of electic fields we have discussed the foce on chages due to fields. Analogously, we have found the magnetic foce on moving chages and cuents. Ou discussion of electic fields is moe complete, howeve, since we have ways to calculate electic fields due to souces, electic chages. We now need to addess the souces of magnetic fields. The BiotSavat Law will be intoduced as the analog of the Coulomb's Law integals ove continuous distibutions to get electic fields. In cases of symmety we could use Gauss's Law to find electic fields; as the magnetic analog of this we will intoduce Ampee's law. Foce on Q Ho IL Electic Fields F = Q E Magnetic Fields F = Q v äb F = I Ÿ s äb
4 4 Chapte F  Magnetism Field due to Q Ho IL ` E = k e q ò E ÿ A = ε 0 Q enclosed Biot Savat Law Ampee' s Law Coulomb's Law and Gauss's Law ae mathematically equivalent fo electostatics. Similaly, we will see that the BiotSavat Law is equivalent to Ampee's Law fo magnetostatics. Electostatics allows no movement and thus no cuents. Magnetostatics allows cuents but equies all cuents to be steady. Gauss's Law is fully coect even beyond electostatics and is one of Maxwell's equations. Ampee's Law, as discussed in this chapte, is only coect in the context of magnetostatics. Next chapte we will intoduce Maxwell's addition to Ampee's Law; this will make it geneally coect and it will become one of Maxwell's equations. F.4  BiotSavat Law E = The BiotSavat law elates the magnetic field at some point P to the cuent in a wie. The analogous expession fo electic fields is 4 p ε 0 ` Savat law is q. The souce is a cuent I though an infinitesimal segment of wie s. Take the vecto to be fom the souce to P. The Biot B = m 0 4 p I s ä `. I P s Deivation Using a Test Pole F PW g 0 I B B WP s F WP Exploiting duality symmety we can deive the BiotSavat Law. To do this intoduce a test magnetic pole of stength g 0 at the position P. The fact that these poles have neve been obseved need not distub us. The field at the wie due to the pole is B WP = m 0 ` 4 p g 0. The negative sign is thee because the vecto in the diagam is pointing towad the pole whee we usually take as pointing away fom the chage o pole. The foce on the wie due to the pole is then F WP = I s äb WP =  m 0 4 p g 0 I s ä `. Using Newton's thid law we can elate this to the foce on the pole due to the wie
5 Chapte F  Magnetism 5 F PW =  F WP = m 0 4 p g 0 I s ä `. The foce on a magnetic pole g 0 in a field B is F = g 0 B so using the pole as a test pole we can wite the field at P due to the wie as B = F PW = m 0 g 0 4 p I s ä `, which is just the BiotSavat Law. Note that the esult is independent of ou test pole. Examples Field at the cente of a cicula ac I s q P Conside a cicula ac of adius R and of angle q in the xyplane with a counteclockwise cuent. The vecto is fom s to the oigin, which is the point P. B = m 0 4 p I s ä` Using A äb = A B sin q ù we get Fo evey point on the ac we have = R = const. giving: The integal is just the total ac length Ÿ s = R q giving s ä` = s ÿ ÿ z`. B = z` m0 I 4 p B = z` R s. m 0 I 4 p R q. Field at a pependicula distance z 0 fom the cente of a cicle
6 6 Chapte F  Magnetism z z 0 P 0 I x R q Now conside a full cicle of adius R in the xyplane with the cente at the oigin and a counteclockwise cuent. Choose the point P to be at z 0 along the positive zaxis. Integate aound the cicle by vaying q fom 0 to p. The position as a function of q is given by the vecto. s = XR cos q, R sin q, 0\ The s is the infinitesimal change in this vecto unde an infinitesimal change in angle, q. s = = XR sin q q, R cos q q, 0\ The vecto is the vecto fom s, which is at, to P which is at 0 = X0, 0, z 0 \. This gives By the Pythagoean theoem, the magnitude of is just Using ` = we get an expession fo the field. 3 = 0  = XR cos q, R sin q, z 0 \. = R + z 0. B = m 0 I s ä 4 p 3 The coss poduct can now be explicitly evaluated using the deteminant method s ä = Since is a constant we can bing the tem out of the integal giving 3 x` ỳ z` R sin q q R cos q q 0 R cos q R sin q z 0 = x` Hz 0 R cos q q  0L ỳ Hz 0 R sin q q  0L +z` IR sin q q  R cos q qm = Yz 0 R cos q, z 0 R sin q, R ] q y This gives thee simple integals. B = m 0 I 4 p HR + z 0 L 3ê 0 p Yz 0 R cos q, z 0 R sin q, R ] q. 0 p z 0 R cos q q = 0
7 Chapte F  Magnetism 7 0 pz 0 R sin q q = 0 0 p R q = p R The final esult can, finally, be witten B = z` m0 I R HR + z 0 L 3ê. Field due to a line segment y a P f x s x x+ x I x x Fo some staightline segment choose the xdiection to be the diection of the cuent and take the segment to be between x and x. The point P is on the yaxis at y = a, whee x = 0 is the point on the line closest to P. The vecto s is the vecto fom x to x + x And the vecto is fom the s to P. s = x` x = x x` + a ỳ The magnetic field at P is Evaluating the coss poduct B = m 0 I s ä 4 p 3 and using = x + a gives s ä = x` x ä Hx x` + a ỳl = z` a x B = z` a m 0 I 4 p x This can be evaluated using a tig substitution. Define the angle f as shown. The substitution is x x = a tan f. x Hx + a L 3ê. The diffeential becomes and becomes a x = cos f f Define f and f as the f values coesponding to x and x = x + a = a cos f ï = Hx + a L 3ê cos 3 f a 3
8 8 Chapte F  Magnetism This gives the final esult B = z` a m 0 I f cos 3 f 4 p f a 3 a cos f f = z` m0 I f cos 4 p a f f. f This can also be witten in tems of the oiginal x vaiables B = z` m0 I 4 p a Hsin f  sin f L. B = z` m0 I 4 p a x x + a  x x + a. Note that with the choice of f given hee that a negative x value coesponds to a negative f value. Field of a long staight wie An impotant special case of this is that of a long staight wie. The field magnitude a distance fom a long staight wie becomes x Ø  and x Ø ï f Ø  p = 90 and f Ø p = 90 Hsin f  sin f L Ø  HL =. B = m 0 I p. To get the diection of the field fo a long staight wie, o fo that matte fo a segment, put the thumb of you ight hand in the diection of the cuent. The field ciculates aound the wie in the diection given by you finges. F.5  Magnetic Foces, m 0 and the Definition of the Coulomb We have not yet assigned a value to the constant m 0. We also haven't given a definition of the Coulomb. The Coulomb will be defined in tems of the Ampee, C = A ÿs, and then the Ampee's definition will be established when we assign a value to the constant m 0. To gain a physical undestanding of these definitions we will conside the magnetic foce between paallel wies and deive a magnetic analog to Coulomb's Law. Foces between Paallel Wies { B B I F I a Conside a long wie with cuent I and a paallel segment of length { a distance a fom the long wie. The paallel segment will have a cuent I ; its cuent will be supplied by pependicula wies coming fom infinity. The magnetic foces on these pependicula segments will cancel and the net foce will just be the foce on the segment. To find this foce takes two steps: Define B to be the field due to I at I and then define F as the foce on I due to B. B = m 0 I p a The diection of B is out of the page. We can then find the magnetic foce. The diection of this foce is towad the othe wie and its magnitude is F = I { äb
9 Chapte F  Magnetism 9 F = I { B = m 0 p I I { a. We can make a geneal statement about magnetic foces between cuents. Paallel cuents attact and antipaallel cuents epel. m 0 and the Ampee In the expession fo the foce both { and a ae lengths; it follows that the SI units of m 0 ae NêA. It should now be clea that edefining an Ampee will change the numeical value of m 0 ; assigning a value to it will then povide a definition of the Ampee. The constant m 0 could be emoved completely by defining its value to be but fo histoical easons we choose diffeently. The value of m 0 is m 0 = 4 p µ07 N A. If an expeiment wee set up with the aangement above, then it could be used to explicitly calibate an ammete. If the two cuents ae foced to be equal I = I = I, then measuing the foce and using the expeiment's values fo { and a would give the cuent in Ampees. F.6  Ampee's Law Ampee's law is mathematically equivalent to the BiotSavat law fo magnetostatics, whee all cuents ae steady giving constant fields. This equivalence cannot be demonstated at this level. We will use Ampee's law similaly to Gauss's law. In cases of symmety we will use it to find magnetic fields fom cuents. Ampee's law is B = m 0 I enclosed. The integal is aound a closed contou and I enclosed is the total cuent enclosed by that contou. If the integal is ove some closed contou then thee ae many diffeent sufaces (an infinite numbe) that have that contou as its bounday. An example is the Eath's equato that has the nothen hemisphee, the southen hemisphee and a disk though the Eath's cente as diffeent sufaces that shae it as thei boundaies. The cuent I is the cuent piecing any suface that has the contou as its bounday. We can elate the oientation of the bounday to the oientation of the contou (the diection of integation aound the contou.) Cylindical Symmety In any case of cylindical symmety choose a cicula contou. By symmety and using Gauss's law fo magnetism we get that the field otates aound the axis. This gives B = p B. Inseting this into Ampee's law gives the geneal expession fo cylindical symmety Note that the field fo a long wie is a tivial special case of this. B = m 0 I enclosed. p Long Solenoid
10 0 Chapte F  Magnetism { 4 3 I B Fo an (infinitely) long solenoid take the cuent to be I and the density of tuns to be n. # of tuns n = length The field inside the solenoid is unifom and the field outside is zeo. (The field outside appoaches zeo as the length become infinite.) Choose the contou to be fou segments as shown Thee ae n { tuns though the contou giving B = B + B + 3 = B { I enclosed = n { I It follows fom Ampee's law that the field anywhee inside a long solenoid is B = m 0 n I. B + 4 B F.7  Cuent Loops as Magnetic Dipoles The net foce on a cuent loop in a unifom magnetic field is zeo. The field does affect the loop, though. Thee is a toque on it. F net = I s äb = I K s O äb = 0 We saw ealie using electomagnetic duality (the analogy between electic and magnetic fields and chages) that the toque on a magnetic dipole is t = m äb. In showing thee is a toque on a cuent loop we will demonstate that a cuent loop is a magnetic dipole. This is a second souce of magnetic dipoles; in addition to pemanent magnets being dipoles, we will now see that cuent loops ae dipoles as well. To calculate the dipole moment of a loop o Ntun coil, we will fist find the moment of a single tiangula loop by calculating the toque on a tiangula loop in a unifom field. A Single Tiangula Loop The definition of toque about an oigin due to some foce is t = äf, whee is the vecto fom the oigin to whee the foce F acts. Conside fist a line segment of length { with cuent I. In finding the foce on this segment we took { to be in the diection of the cuent and the foce is F = I { äb. The toque on the segment becomes that of the foce acting at the midpoint of the segment mid t = I mid äi{ äbm. Conside a single tiangula loop caying a cuent I in a unifom magnetic field B. Toque depends on one's choice of oigin, but wheneve the net foce vanishes the net toque is independent of the choice of oigin. We will choose the oigin to be at the cente of one side; this
11 Chapte F  Magnetism the net foce vanishes the net toque is independent of the choice of oigin. We will choose the oigin to be at the cente of one side; this emoves the contibution of that segment to the toque since mid = 0. The two sides that do contibute ae labeled { and {, and take thei diections to be the diection of the cuent. { {  { { t = I  { We can simplify this using an identity satisfied by coss poducts. äi{ äbm + I { äi{ äbm + 0 A äib äcm + B äic ä AM + C äia äbm = 0 This identity can be ewitten as A bit of algebaic manipulation A Ø {, B Ø { and C Ø B gives A äib äcm  B äia äcm = IA äbm äc. t = I { ä { äb. Recall that the magnitude of the coss poduct is the aea of a paallelogam. A tiangle is half of that. The diection of the coss poduct is pependicula to the two vectos and thus to the tiangle. It follows that the aea vecto of the tiangle is A = { ä {. Note that the ighthand ule gives the outwad nomal as the diection of A and the ighthand ule also associates that diection with a counteclockwise cuent. We can now wite toque as t = I A äb. Since the toque on a magnetic dipole is t = m äb, we can wite the magnetic moment of a single tiangula cuent loop as Plana Cuent Loops o Coils with N Tuns m = I A. We can now genealize this esult to a geneal plana loop. Any loop may be boken up into infinitesimal pieces each of aea A. Summing ove all these infinitesimal aeas gives A = Ÿ A. The aea vecto has a magnitude A and its diection is nomal to the loop. A = A ǹ To get the pope diection use a ighthand ule. Wap you finges aound loop in the diection of the cuent. The thumb of you ight hand points in the diection of the unit nomal ǹ that gives the diection of A. The magnetic moment is then m = I A. HSingle tun loopl If thee ae N tuns then each tun contibutes I A to the magnetic moment. The total magnetic dipole moment is m = N I A. HN tun coill N is the numbe of tuns, I is the cuent and A = A ǹ is the aea vecto defined as above.
Forces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationMagnetic Field in a TimeDependent Capacitor
Magnetic Field in a TimeDependent Capacito 1 Poblem Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 8544 (Octobe 3, 23) Reconside the classic example of the use of Maxwell s displacement
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationGauss s law relates to total electric flux through a closed surface to the total enclosed charge.
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationNotes on Electric Fields of Continuous Charge Distributions
Notes on Electic Fields of Continuous Chage Distibutions Fo discete pointlike electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationEXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD
260 161. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationPoynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions
Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallelplate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationSELFINDUCTANCE AND INDUCTORS
MISN0144 SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. SelfInductance L.........................................
More informationA couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= F A
5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (=  ) Since the foces ae
More informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationLesson 8 Ampère s Law and Differential Operators
Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationPhysics 505 Homework No. 5 Solutions S51. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.
Physics 55 Homewok No. 5 s S5. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More information1.4 Phase Line and Bifurcation Diag
Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) YanBin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More information92.131 Calculus 1 Optimization Problems
9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
More informationSolutions for Physics 1301 Course Review (Problems 10 through 18)
Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal
More information2. SCALARS, VECTORS, TENSORS, AND DYADS
2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that
More informationELECTRIC CHARGES AND FIELDS
Chapte One ELECTRIC CHARGES AND FIELDS 1.1 INTRODUCTION All of us have the expeience of seeing a spak o heaing a cackle when we take off ou synthetic clothes o sweate, paticulaly in dy weathe. This is
More informationTECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications
JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy
More informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More information10. Collisions. Before During After
10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: F(t)
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More information