Outside the solenoid, the field lines are spread apart, and at any given distance from the axis, the field is weak.

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Emil Gaines
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1 Applications of Ampere s Law continued. 2. Field of a solenoid. A solenoid can have many (thousands) of turns, and perhaps many layers of windings. The figure shows a simple solenoid with just a few windings and a spacing which is exaggerated. Fig The higher the ratio of length to diameter of the solenoid, and the tighter the windings, the more uniform is the magnetic field inside, and the less leakage there is of field lines before the ends of the solenoid. Outside the solenoid, the field lines are spread apart, and at any given distance from the axis, the field is weak. How do we calculate the magnetic field at the centre of a long, narrow, tightly wound solenoid? Use Ampere s law. This would not be an easy problem to do without Ampere s law. PHYS W 1

2 Ampere s law involves integrating around a closed loop. In the figure, abcd is such a loop for the long solenoid. Along the side ab, is uniform and constant. b a dl L Along the sides bc and ad, d l 0 because. l d Along the side cd, because. dl dl 0 c L If n is the number of turns per unit length, then 0 Fig µ 0 I enc µ 0 nli PHYS W 2

As length/diameter ratio increases, the field along the axis inside is more uniform. 3.

3 µ ni 0 (solenoid) Note in the figure the plot of vs. distance along the axis the field value at the ends is about ½ that at the centre. As length/diameter ratio increases, the field along the axis inside is more uniform. 3. Field of a toroid A toroidal solenoid is a closed loop, usually of some magnetic material, with windings all the way around it. Fig If the windings are tight, they are equivalent to adjacent circular loops, and the resulting field will be concentric with the toroid s axis. Fig PHYS W 3

4 In the figure, the only path that encloses current is path 2. Path 1 encloses no current. 0 for path 1. 0 Path 3 encloses no net current. for path 3. There will be an enclosed current for path 2, in between the inner and outer circumferences. For this path, we expect the field to be uniform and constant. dl N 2πr If is the total number of windings, 2 π r µ 0NI I enc NI µ 0NI µ ni where n is the number of turns per unit length. 2πr 0 This equation is strictly correct only for the core inside the windings being a vacuum. ( µ 0 is the permeability of free space). However, there is very little difference if the core is air, or a solid non-magnetic material. If the core Is a magnetic material, there can be a big change in over the value in free space. PHYS W 4

5 ohr magneton As discussed earlier, materials that are magnetic have their magnetic origins at the atomic level. Electrostatic phenomena do not require moving charges. Magnetic phenomena do. For a magnetized piece of material, say a magnet, the moving charges can be viewed as electron spins, or electron orbital motions. oth of these can be fully understood only by studying quantum mechanics. ut a classical physics analysis can still be useful for some understanding. The electron spin has no classical analogue but orbital motion does. The figure shows a classical picture of an electron in an orbit of radius r, and speed v. Note the orbital angular momentum and magnetic moment are oppositely directed. (the same applies to the electron spin). 2 Recall. µ IA Iπr µ L Fig PHYS W 5

6 The orbital angular momentum of the electron L mvr e For a charge moving in a circle, the equivalent current is the total charge passing any point in the orbit per unit time, or the charge divided by the period. I µ e T ev 2πr ev 2πr 2 evr ( πr ) 2 L vr L Substituting for, µ m e L µ e 2m Note, and point in opposite directions for an electron. For a positive charge, they would point in the same direction. In the quantum theory of the atom, angular momentum is quantized. It must be an integral multiple of h h 2π e PHYS W 6

7 34 h 6.626x10 J. s 34 h 1.05x 10 J. s µ L So and are more accurately related as µ e 2m L h h e eh h 2m 9.274x10 Am µ is the ohr magneton. e The spin magnetic moment is almost exactly equal to 1 ohr magneton (1.001 µ ) If the magnetic moments of all the atoms or molecules are reasonably well aligned, the material is highly magnetized. PHYS W 7

8 Paramagnetism Paramagnetic materials are those that can be weakly magnetized. In the presence of a magnetic, the material will be weakly attracted. The Enhancement of the field is described by K m, the relative permeability. Here K ranges from to m Diamagnetism Diamagnetic materials are also weakly magnetized, but in the opposite direction to that of an external field. These materials will be repelled by either pole of a magnet. Is typically in the range to Ferromagnetism K m Ferromagnetic materials can be very strongly magnetized. These materials include iron, nickel, and cobalt. Electromagnets use ferromagnetic materials in their cores to increase the value of the magnetic field. In these materials, there exist regions called magnetic domains (see next slide) where strong interactions between atomic magnetic moments cause them to line up parallel to each other. PHYS W 8

9 In the absence of an external field, the domain magnetizations are randomly oriented. When an external field is applied, the domains tend to align themselves parallel to the field. The domain boundaries also shift. The domains that are magnetized in the direction of the applied field grow and those that are magnetized in other directions shrink. Magnetic domains The magnetic field at any point in such a material is greater by K m. For ferromagnetics, K m ,000 When a magnet is brought near to a ferromagnetic material, with a suitable amplifier and speakers, one can hear the noise from the shifting domains. This is called the arkhausen effect. Go to for a good demonstration. PHYS W 9

10 Electromagnetic induction Until now, the only method of producing an emf and having a current flow in a closed loop (a circuit) is to use a battery in the system. However, there is another method which dates back to the experiments of Michael Faraday (and Joseph Henry) in the 1830s. They found in their experiments that a changing magnetic field flux through a circuit will induce a current (and hence an emf). This was an astounding, and unexpected discovery. A battery in a circuit will cause a current to flow There are many ways in which the flux can change. the source of can be moving so that the flux through a closed loop changes the loop can be moving in such a way that the flux through it changes a switch could be closed so that a field is created PHYS W 10 A A If the flux of through the coil, changes, a current will flow. I

11 a switch could be closed so that the loop is a closed circuit and current can flow the coil area could be changed by squeezing it, or by rotating it more turns of wire could be added (or subtracted) to (or from) the coil All of these changes will cause the flux through the coil to change and a current will be observed to flow in the coil (and hence an emf was produced). Only the changes will cause an emf to be produced. The size of the emf produced depends on the rate of change. Review of magnetic flux. For an infinitesimally small patch of area d A, the magnetic flux dφ is given by dφ da da dacosφ Φ da dacosφ If is uniform over a flat area Φ A Acosφ A Fig PHYS W 11

Area is a vector and there are two possible choices for it. Once it is chosen, we must stick with the convention for consistency. A da If and are parallel, is positive.

12 Area is a vector and there are two possible choices for it. Once it is chosen, we must stick with the convention for consistency. A da If and are parallel, is positive. If they are anti-parallel, then the product will be negative. A Note above that when and are parallel (or anti-parallel), the magnitude of the flux is a maximum. A When the flux is zero. Now back to induced emf. PHYS W 12

13 All of the changes that will produce an emf are stated in Faraday s law which is: Faraday s law The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ε dφ dt (Faraday s law of induction) The minus sign means that the emf is induced in such a way as to oppose the change that is causing it. For the diagram on slide 10, with the direction of the current as shown, the flux of must be increasing through the loop. The current then is flowing so that the magnetic field that it produces opposes the change (increasing flux of ) that is causing it. increasing decreasing I I The above diagrams show the direction of the induced current (and hence emf) for flux through the loop that is increasing, and decreasing. PHYS W 13

14 For N turns of wire, ε dφ dt N Note that Faraday s law states two important things: 1. emf is induced only when there is a change in the flux. 2. the magnitude is proportional to the time rate of change of the flux. Lenz s law This law concentrates on the direction of the induced emf and explains the negative sign in the above equation. The statement is: The direction of any magnetic induction effect is such as to oppose the cause of the effect. The cause of the effect is always changing flux. ut this may be due to a varying magnetic field, or motion of the conductors that make up the circuit, or a combination of both. Lenz s law is directly related to energy conservation. If the law didn t apply, we could have positive feedback with flux building up in a runaway process. PHYS W 14

and downward direction. In (a) and (b), the induced emf produces an upward pointing field to oppose the change.

15 Fig The four parts of the above figure show various situations of increasing or decreasing, both in an upward direction, and downward direction. In (a) and (b), the induced emf produces an upward pointing field to oppose the change. In (c) and (d), the induced emf produces a downward pointing field to oppose the change. Examples A simple alternator or ac generator This is a device that generates an emf. You may be familiar with this device from a car. PHYS W 15

16 A rectangular loop is made to rotate with constant angular speed about an axis, as shown. An induced emf is generated. This is the reverse situation of the dc motor that we have studied. ω An alternating emf is generated between terminals a and b. Starting at the stage shown in the figure, as the loop continues to rotate in the direction indicated, the flux increases and an induced emf is generated to oppose this. As the angle o rotates through 180 the flux starts to decrease and the direction of emf changes. PHYS W 16 φ

A In this case, and are uniform and the flux is Φ ε Acos φ d Φ dt Acosωt ωasinωt The magnitude of the emf depends upon (1) rotation speed, (2) the strength of the field, (3) the area of the coil, and

17 A In this case, and are uniform and the flux is Φ ε Acos φ d Φ dt Acosωt ωasinωt The magnitude of the emf depends upon (1) rotation speed, (2) the strength of the field, (3) the area of the coil, and (4) the number of turns, N. Exercise A flat rectangular coil (l x w) is pulled with uniform speed v through a uniform magnetic field with the plane of its area perpendicular to the field. (a) Determine the emf induced in this coil. (b) If the speed and field strength are both tripled, what is the induced emf? PHYS W 17

Displacement Current. Ampere s law in the original form is valid only if any electric fields present are constant in time

Displacement Current. Ampere s law in the original form is valid only if any electric fields present are constant in time Displacement Current Ampere s law in the original form is valid only if any electric fields present are constant in time Maxwell modified the law to include timesaving electric fields Maxwell added an