PERMANENT MAGNETS CHAPTER MAGNETIC POLES AND BAR MAGNETS

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CHAPTER 6 PERAET AGET 6. AGETIC POLE AD BAR AGET We have seen that a small current-loop carrying a current i, prouces a magnetic fiel B o 4 ji ' at an axial point.

1 CHAPTER 6 PERAET AGET 6. AGETIC POLE AD BAR AGET We have seen that a small current-loop carrying a current i, prouces a magnetic fiel B o 4 ji ' at an axial point. Here p ia is the magnetic ipole moment of the current loop. The vector A represents the area-vector of the current loop. Also, a current loop place in a magnetic fiel B experiences a torque r (Tx. (i)... (ii) We also know that an electric ipole prouces an electric fiel E 4rce at an axial point an it experiences a torque (iii) rpx... (iv) when place in an electric fiel. Equations (i) an (ii) for a current loop are similar in structure to the equations (iii) an (iv) for an electric ipole with p taking the role of p an -y - taking the role of 4 it 4nc The similarity suggests that the behaviour of a current loop can be escribe by the following hypothetical moel: (a) There are two types of magnetic charges, positive magnetic charge an negative magnetic charge. A magnetic charge m place in a magnetic fiel B experiences a force F mb.... (6.) The force on a positive magnetic charge is along the fiel an the force on a negative magnetic charge is opposite to the fiel. (b) A magnetic charge m prouces a magnetic fiel Po m B 4 it (6.) at a istance r from it. The fiel-is raially outwar if the magnetic charge is positive an is inwar if it is negative. (c) A magnetic ipole is forme when a negative magnetic charge - m an a positive magnetic charge + m are place at a small separation. The magnetic ipole moment is p m an its irection is from - m to + m. The line joining m an + m is calle the axis of the ipole. () A current loop of area A carrying a current i may be replace by a magnetic ipole of ipole moment (j m ia place along the axis of the loop. The area-vector A points in the irection - m to + in. current loop + m equivalent magnetic ipole Figure 6. The moel is very useful in stuying magnetic effects an is wiely use. It is customary to call a positive magnetic charge a north pole an a negative magnetic charge a south pole. They are represente by the letters an respectively. The quantity m is calle pole strength. From the equation m ia or F mb, we can easily see that the unit of pole strength is A-m. We can fin the magnetic fiel ue to a magnetic ipole at any point P using equation (6.) for both the poles. A solenoi very closely resembles a combination of circular loops place sie by sie. If i be the current through it an A be the area of cross-section, the ipole moment of each turn is p ia. In our moel, each turn may be replace by a small ipole place at the centre I l (a) (c) Figure 6. s K i--i (b)

2 Permanent agnets 75 or, or, or, v 6 y 4 _B B, J_ / (B IB) K ' / + B -5 or, B -5 x 4 nt x T. The oscillating magnet is in en-on position of the short magnet. Thus, the fiel B can be written as Ho ' B 4it ' or, ' -- B o We have, /--g(l +6 ) - ' [(7 x -y + ( x - V] kg-m ; - f xl - 5 kg-m. T-n r n y mb,, 4k / 47t x 5 x 6 kg/m or, ;; B T 6 x(5x T)x^-s - 7 A-m. (b) In this case the moment of inertia becomes /' - ~ (L + b' ) where b' '5 cm. (i) - -5 X 7 - A-m' T-m x ( x "8 T) x ( x - x m) The time perio woul be (ii) 9. A bar magnet of mass g, length 7' cm, with l'o cm an height '5 cm takes k/ secons to complete an oscillation in an oscillation magnetometer place in a horizontal magnetic fiel of 5 pt. (a) Fin the magnetic moment of the magnet, (b) If the magnet is put in the magnetometer with its '5 cm ege horizontal, what woul be the time perio? olution : (a) The moment of inertia of the magnet about the axis of rotation is II Diviing T by equation (i), " rr r - y j y (L + b' ) y (7 cm) + (-5 cm) 5 '99 / -99 x k or, T s 496tc s. (7 cm) + (- cm)' QUETIO FOR HORT AWER. Can we have a single north pole? A single south pole?. Do two istinct poles actually exist at two nearby points in a magnetic ipole?. An iron neele is attracte to the ens of a bar magnet but not to the mile region of the magnet. Is the material making up the ens of a bar magnet ifferent from that of the mile region? 4. Compare the irection of the magnetic fiel insie a solenoi with that of the fiel there if the solenoi is replace by its equivalent combination of north pole an south pole. 5. ketch the magnetic fiel lines for a current-carrying circular loop near its contre. Replace the loop by an equivalent magnetic ipole an sketch the magnetic fiel lines near the centre of the ipole. Ientify the ifference. 6. The force on a north pole, F mb, is parallel to the fiel B. Does it contraict our earlier knowlege that a magnetic fiel can exert forces only perpenicular to itself? 7. Two bar magnets are place close to each other with their opposite poles facing each other. In absence of other forces, the magnets are pulle towars each other an their kinetic energy increases. Does it contraict our earlier knowlege that magnetic forces cannot o any work an hence cannot increase kinetic energy of a system? 8. agnetic scalar potential is efine as U(r) - U{r,) - - J B i r i Apply this equation to a close curve enclosing a long

76 Concepts of Physics straight wire. The RH of the above equation is then - p by Ampere's law. We see that U(r) * U(rt) even when r r,. Can we have a magnetic scalar potential in this case? 9.

3 76 Concepts of Physics straight wire. The RH of the above equation is then - p by Ampere's law. We see that U(r) * U(rt) even when r r,. Can we have a magnetic scalar potential in this case? 9. Can the earth's magnetic fiel be vertical at a place? What will happen to a freely suspene magnet at such a place? What is the value of ip here?. Can the ip at a place be zero? 9?. The reuction factor if of a tangent galvanometer is written on the instrument. The manual says that the current is obtaine by multiplying this factor to tan6. The proceure works well at Bhuwaneshwar. Will the proceure work if the instrument is taken to epal? If there is some error, can it be correcte by correcting the manual or the instrument will have to be taken back to the factory? OBJECTIVE I 4. A circular loop carrying a current is replace by an equivalent magnetic ipole. A point on the axis of the loop is in (a) en-on position (b) broasie-on position (c) both () none. A circular loop carrying a current is replace by en equivalent magnetic ipole. A point on the loop i in (a) en-on position (b) broasie-on position (c) both () none. When a current in a circular loop is equivalently replace by a magnetic ipole, (a) the pole strength m of each pole is fixe (b) the istance between the poles is fixe (c) the prouct m is fixe () none of the above. Let r be the istance of a point on the axis of a bar magnet from its centre. The magnetic fiel at such a point is proportional to oo (b) AT (c) j () none of these. r 5. Let r be the istance of a point on the axis of a magnetic ipole from its centre. The magnetic fiel at such a point is proportional to i (b) ~ _ (c) () none of these. r 6. Two short magnets of equal ipole moments are fastene perpenicuarly at their centres (figure 6-Q). The magnitue of the magnetic fiel at a istance from the centre on the bisector of the right angle is (a) 4n (b) o / 4k (c).c 7. agnetic meriian is (a) a point (c) a horizontal plane Po / 4K ()J^- 4n,'P -' Figure 6-Q (b) a line along north-south () a vertical plane. 8. A compass neele which is allowe to move in a horizontal plane is taken to a geomagnetic pole. It (a) will stay in north-south irection only (b) will stay in east-west irection only (c) will become rigi showing no movement () will stay in any position. 9. A ip circle is taken to geomagnetic equator. The neele is allowe to move in a vertical plane perpenicular to the magnetic meriian. The neele will stay (a) in horizontal irection only (b) in vertical irection only (c) in any irection except vertical an horizontal () in any irection it is release.. Which of the following four graphs may best represent the current-eflection relation in a tangent galvanometer? Figure 6-Q. A tangent galvanometer is connecte irectly to an ieal battery. If the number of turns in the coil is ouble, the eflection will (a) increase (b) ecrease (c) remain unchange () either increase or ecrease.. If the current is ouble, the eflection is also ouble in > m (a) a tangent galvanometer (b) a moving-coil galvanometer (c) both () none.. A very long bar magnet is place with its north pole coinciing with the centre of a circular loop carrying an electric current i. The magnetic fiel ue to the magnet at a point on the periphery of the wire is B. The raius of the loop is a. The force on the wire is (a) very nearly naib perpenicular to the plane of the wire (b) naib in the plane of the wire (c) 7laiB along the magnet () zero.

Permanent agnets 77 OBJECTIVE II. Pick the correct options. (a) agnetic fiel is prouce by electric charges only. (b) agnetic poles are only mathematical assumptions having no real existence.

4 Permanent agnets 77 OBJECTIVE II. Pick the correct options. (a) agnetic fiel is prouce by electric charges only. (b) agnetic poles are only mathematical assumptions having no real existence. (c) A north pole is equivalent to a clockwise current an a south pole is equivalent to an anticlockwise current. () A bar magnet is equivalent to a long, straight current.. A horizontal circular loop carries a current that looks clockwise when viewe from above. It is replace by an equivalent magnetic ipole consisting of a south pole an a north pole. (a) The line shoul be along a iameter of the loop. (b) The line shoul be perpenicular to the plane of the loop. (c) The south pole shoul be below the loop. () The north pole shoul be below the loop.. Consier a magnetic ipole kept in the north-south irection. Let P,, P Q Q be four points at the same istance from the ipole towars north, south, east an west of the ipole respectively. The irections of the magnetic fiel ue to the ipole are the same at (a) P, an P (b) Q, an Qt (c) P, an Q, () P an Q,, 4. Consier the situation of the previous problem. The irections of the magnetic fiel ue to the ipole are opposite at (a) P, an P (b) Q, an Q., (c) P, an Q, () P an Q 5. To measure the magnetic moment of a bar magnet, one may use (a) a tangent galvanometer (b) a eflection galvanometer if the earth's horizontal fiel is known (c) an oscillation magnetometer if the earth's horizontal fiel is known () both eflection an oscillation magnetometer if the earth's horizontal fiel is not known. EXERCIE. A long bar magnet has a pole strength of A-m. Fin the magnetic fiel at a point on the axis of the magnet at a istance of 5 cm from the north pole of the magnet.. Two long bar magnets are place with their axes coinciing in such a way that the north pole of the first magnet is - cm from the south pole of the secon. If both the magnets have a pole strength of A-m, fin the force exerte by one magnet on the other.. A uniform magnetic fiel of " x ~ T exists in the space. Fin the change in the magnetic scalar potential as one moves through 5 cm along the fiel. 4. Figure (6-E) shows some of the equipotential surfaces of the magnetic scalar potential. Fin the magnetic fiel B at a point in the region. Figure 6-E 5. The magnetic fiel at a point, cm away from a magnetic ipole, is foun to be ' x T. Fin the magnetic moment of the ipole if the point is (a) in en-on position of the ipole an (b) in broasie-on position of the ipole. / 6. how that the magnetic fiel at a point ue to a magnetic ipole is perpenicular to the magnetic axis if the line joining the point with the centre of the ipole makes an angle of tan ~ '(/) with the magnetic axis. 7. A bar magnet has a length of 8 cm. The magnetic fiel at a point, at a istance cm from the centre in the broasie-on position is foun to be 4 x " T. Fin the pole strength of the magnet. 8. A magnetic ipole of magnetic moment '44 A-m " is place horizontally with the north pole pointing towars north. Fin the position of the neutral point if the horizontal component of the earth's magnetic fiel is 8 mt. 9. A magnetic ipole of magnetic moment '7 A-m is place horizontally with the north pole pointing towars south. Fin the position of the neutral point if the horizontal component of the earth's magnetic fiel is 8 jut.. A magnetic ipole of magnetic moment '7/ A-m " is place horizontally with the north pole pointing towars east. Fin the position of the neutral point if the horizontal component of the earth's magnetic fiel is 8 T.. The magnetic moment of the assume ipole at the earth's centre is 8' x A-m. Calculate the magnetic fiel B at the geomagnetic poles of the earth. Raius of the earth is 64 km.. If the earth's magnetic fiel has a magnitue '4 x 5 T at the magnetic equator of the earth, what woul be its value at the earth's geomagnetic poles?. The magnetic fiel ue to the earth has a horizontal component of 6 (jt at a place where the ip is 6. Fin the vertical component an the magnitue of the fiel. 4. A magnetic neele is free to rotate in a vertical plane which makes an angle of. 6 with the magnetic meriian. It' the neele stays in a irection making an angle of tan with the horizontal, what woul be the ip at that place?

78 Concepts of Physics 5. The neele of a ip circle shows an apparent ip of 45 in a particular position an 5 when the circle is rotate through 9. Fin the true ip. 6.

If the horizontal component of the earth's magnetic fiel is '6 x " 5 T an raius of the coil is cm, fin the number of turns in the coil. 7. A moving-coil galvanometer has a 5-turn coil of size cm x cm.

5 78 Concepts of Physics 5. The neele of a ip circle shows an apparent ip of 45 in a particular position an 5 when the circle is rotate through 9. Fin the true ip. 6. A tangent galvanometer shows a eflection of 45 when ma of current is passe through it. If the horizontal component of the earth's magnetic fiel is '6 x " 5 T an raius of the coil is cm, fin the number of turns in the coil. 7. A moving-coil galvanometer has a 5-turn coil of size cm x cm. It is suspene between the magnetic poles proucing a magnetic fiel of 5 T. Fin the torque on the coil ue to the magnetic fiel when a current of ma passes through it. 8. A short magnet prouces a eflection of 7 in a eflection magnetometer in Tan-A position when place at a separation of cm from the neele. Fin the ratio of the magnetic moment of the magnet to the earth's horizontal magnetic fiel. 9. The magnetometer of the previous problem is use with the same magnet in Tan-B position. Where shoul the magnet be place to prouce a 7 eflection of the neele?. A eflection magnetometer is place with its arms in north-south irection. How an where shoul a short magnet having /B 4 A-m /T be place so that the neele can stay in any position?. A bar magnet takes it/ secon to complete one oscillation in an oscillation magnetometer. The moment of inertia of the magnet about the axis of rotation is ' x " 4 kg-m" an the earth's horizontal magnetic fiel is pt. Fin the magnetic moment of the magnet.. The combination of two bar magnets makes oscillations per secon in an oscillation magnetometer when like poles are tie together an oscillations per secon when unlike poles are tie together. Fin the ratio of the magnetic moments of the magnets. eglect any inuce magnetism.. A short magnet oscillates in an oscillation magnetometer with a time perio of ' s where the earth's horizontal magnetic fiel is 4 pt. A ownwar current of 8 A is establishe in a vertical wire place cm east of the magnet. Fin the new time perio. 4. A bar magnet makes 4 oscillations per minute in an oscillation magnetometer. An ientical magnet is emagnetize completely an is place over the magnet in the magnetometer. Fin the time taken for 4 oscillations by this combination. eglect any inuce magnetism. 5. A short magnet makes 4 oscillations per minute when use in an oscillation magnetometer at a place where the earth's horizontal magnetic fiel is 5 pt. Another short magnet of magnetic moment '6 A-m is place cm east of the oscillating magnet. Fin the new frequency of oscillation if the magnet has its north pole (a) towars, north an (b) towars south. AWER - (a) 7. (). (a). (a), (b) 4. (c), (). 4 x " 4 T. (b) 8. (). -5 x " OBJECTIVE I. (c) 9. () 4. (). (c) OBJECTIVE II. (b), () 5. (b), (c), () EXERCIE. ecreases by ' x ~ T-m 4. x - 4 T 5. (c). (c). (a), (b) 6. (c). (b) 9. cm south of the ipole. cm from the ipole, tan" '\' south of east. 6 pt. 6-8 x " 5 T. 45 pt, 5 pt x " 4 -m A x 9. 7"9 cm from the centre. - cm from the neele, north pole pointing towars south. 6 A-m. : 5. (a) - A-m an (b) * A-m 7. 6 x " 5 A-m 8. at a istance of cm in the plane bisecting the ipole. 76 s 4. ^ minutes 5. (a) 8 oscillations/minute (b) 54 oscillations/minute

6 CHAPTER 6 PERAET AGET. m A-m, 5 cm.5 m m B r. m m A-m r cm. m we know Tesla Force exerte by tow magnetic poles on each other m m r 4.5 v. B v B l..5. T-m ince the sigh is ve therefore potential ecreases. 4. Here x sin cm 5 cm V B x. 5 T m m ince B is perpenicular to equipotential surface. Here it is at angle with (+ve) x-axis an B 4 T 5. B 4 T cm. m (a) if the point at en-on postion. 7 B 4 ( ) Am (b) If the point is at broa-on position 6. Given : 7 4 ( Am ) tan tan tan tan tan cot cot V. 4 T-m. 4 T-m. 4 T-m.4 4 T-m X In cm P tan We know tan Comparing we get, tan cot or, tan tan(9 ) or 9 or + 9 Hence magnetic fiel ue to the ipole is r to the magnetic axis. 7. agnetic fiel at the broa sie on position : B / l 8 cm cm 4 6 m 4 m 8 / A-m 9 m 8 / / 5 6.

7 6. Permanent agnets 8. We know for a magnetic ipole with its north pointing the north, the neutral point in the broasie on position. Again B in this case BH ue to earth.44 8 T m cm In the plane bisecting the ipole. 9. When the magnet is such that its orth faces the geographic south of earth. The neutral point lies along the axial line of the magnet / m cm. agnetic moment.7 A-m B T m cm. The geomagnetic pole is at the en on position of the earth. 8 B (64 ). B.4 5 T Given.4 5 R.4 5 R.4 R B at Poles T R. (ip) 6 B cos 6 B T 6 6 T 6 T B V B sin T 45 T If an be the apparent ips shown by the ip circle in the r positions, the true ip is given by Cot Cot + Cot Cot Cot 45 + Cot 5 Cot W -6 tan B P E

8 5. We know in r Give :.6 5 T 45 i ma A tan n? r cm. m 5 BH tan r.6 n turns i 6. n 5 A cm cm 4 m i A B.5 T ni A B niab in Given 7 cm. m We know ( ) tan tan [As the magnet is short] 4 Permanent agnets (.) tan A-m T 8. (foun in the previous problem).75 A-m T 7,? 4 / ( ) tan l << neglecting l w.r.t. Tan m 7.9 cm 9. Given 4 A-m /T ince the magnet is short l can be neglecte o, m cm with the northpole pointing towars south.. Accoring to oscillation magnetometer, T A-m 6 A-m 6 6.

9 Permanent agnets mb. We know : H For like poles tie together For unlike poles T T. B B wire.4 6 o i r T. T. T T 5 T B T BH T 4 Here T.76 4 (4 ) T 4 min T? T T 4T T T.56 min 6T 8 For oscillation Time taken.56 min. For 4 Oscillation Time oscillations/minute 5 T m of secon magnet.6 A-m cm. m (a) For north facing north B m.6 T 8 B 4 B B H (b) For north pole facing south B 4 B B H B B H B B H 5 45 min osci/min osci/min

10. Magnetism. ) it is. S G appropriate to call the magnetic pole

10. Magnetism. ) it is. S G appropriate to call the magnetic pole 10 agnetism The wor magnetism is erive from iron ore magnetite (Fe 3 O 4, which was foun in the islan of magnesia in Greece It is believe that the Chinese ha known the property of the magnet even in 000