# k u (t) = k b + k s t

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 2 Stripe domains in thin films 2.1 Films with perpendicular anisotropy In the first part of this chapter, we discuss the magnetization of films with perpendicular uniaxial anisotropy. The easy axis of the anisotropy is chosen parallel to the film normal and favors an out-of-plane magnetization. The tendency of the magnetization to rotate out-of-plane, however, is hindered by the presence of the magnetic charges on the surface of the film which contribute to the magnetostatic energy. As a consequence, the magnetostatic energy (or the shape anisotropy) is smallest when the magnetization lies in the plane of the film. The competition between uniaxial and shape anisotropy determines whether the magnetization tends to be perpendicular or parallel to the film normal, provided that the magnetization is uniform. The two quantities are compared by means of the quality factor [29] Q = 2k u (2.1) µ 0 Ms 2 the ratio between the uniaxial and the shape anisotropy. The uniaxial anisotropy is a function of the thickness t of the film and it is given by the sum of the volume anisotropy constant k b and of the surface anisotropy constant k s divided by t: k u (t) = k b + k s t (2.2) If Q < 1, the shape anisotropy is dominant and the magnetization tends to lie in-plane. If Q > 1, the uniaxial anisotropy forces the anisotropy to lie out-of-plane. So far we have seen that the tendency of the magnetization to lie in or out-of-plane is connected to the ratio of uniaxial and shape anisotropies. To understand whether the system shows magnetic domains or a single domain, we have to analyse the energy of these configurations. In general, magnetic domains occur when the gain in magnetostatic energy due to the domain structure is bigger than the energy required to form the domain walls. 15

2 16 Chapter 2. Stripe domains in thin films 2.2 Domains separated by walls of negligible width In order to calculate the total energy of a stripe configuration, some studies [38, 40, 42] follow the pioneering work of Kittel [39] who considered a series of stripes of width D, with the magnetization pointing alternatively up and down (Fig. 2.1). The stripes are assumed uniformly magnetised and separated by a wall of width δ w much smaller than the domains D. Figure 2.1: Series of stripe domains for a thick film, i.e., for thickness t much bigger than the domain width D. The magnetization is uniform within each stripe and the wall width δ w is negligible with respect to the domain width D. The relevant energy is given by the sum of the magnetostatic energy and of the wall energy. The total wall energy per unit area of the stripe domain configuration in Fig. 2.1 is given by e w = γ w t D (2.3) where γ w = 4 Ak is the energy of a Bloch wall and t/d is the total domain wall area [28]. In the limit of thick films, i.e., t D, the magnetostatic field created by the charges on one surface of the slab does not interact with the field of the other surface. Therefore the value of magnetostatic energy is two times the value obtained for a single surface of the slab. The field created by the surface charges with alternating sign ±M s, is given by the Fourier expansion of a square-wave of amplitude 2πM 2 s. The magnetostatic energy density per unit area for one single surface results [39]: e ms = 0.85M 2 s D (2.4) The dependence of the domain width D from the thickness of the slab t is obtained by minimizing the energy, given by the sum of eq. (2.3) and (2.4), with respect to D. The result is that in the limit of thick films, the domain width grows with the root of the

3 2.2 Domains separated by walls of negligible width 17 thickness: D t. If the thickness of the film is smaller or comparable to the domain width, i.e., t D, the two surfaces interact magnetostatically. The magnetic field H is approximately the same of a film uniformly magnetized, except for regions of size t around the wall Fig. 2.2 [26]. Figure 2.2: Stripe domains in the limit of small thickness, i.e., t D. The stray field crosses the film, side to side, and it is similar to the stray field of a single domain state, uniformly magnetized out-of-plane. The difference is localized around the border of successive stripes; in this region the flux closure slightly reduces the magnetostatic energy with respect to the value of a single domain state. The arrows indicate the stray field direction. In this case the magnetostatic energy is a function of the thickness t and it is found by expanding the magnetic potential in a double Fourier series given by [40, 42]: e ms = 16M 2 s D π 2 n=1 1 (1 exp( 2nπt)) (2.5) 2n D In the limit of thick films, the exponential of eq. (2.5) is negligible and the magnetostatic energy tends to the value given by Kittel, eq. (2.4). In the limit of thin films, i.e., t D, the exponential can be expanded to the first order, so that exp( 2nπt) D =1-2nπt/D and eq. (2.5) becomes: e ms = 2πMs 2 t (2.6) This expression shows that in the limit of thin films, the magnetostatic energy reaches the value of a film of thickness t, uniformly magnetized parallel to the normal. An analytical study of the system in the limit of thin films, has been performed by Kaplan and Gehring [38]. Using the Taylor expansion of t/d up to the second order, they give the expression for the magnetostatic energy: e ms = 2πM 2 s t( t D + 2t πd ln t D ) (2.7) The comparison of this equation with eq. (2.6) shows that the difference in magnetostatic energy between the stripe-domains state and the single domain state is vanishing with the thickness t. The minimization of the total energy, obtained after summation

4 18 Chapter 2. Stripe domains in thin films of equations (2.3) and (2.5), gives the value of the domain width D(t) as a function of the thickness of the film. The numerical solution is plotted in logarithmic scale in Fig. (2.3) [43]. Figure 2.3: Domain width D(t) as a function of t, thickness of the film. The curve is obtained numerically by minimizing the total energy which is given by the sum of equations (2.3) and (2.5) [43]. The axis are normalized to the minimum of the domain width, D m =γ m /1.7 Ms 2. In the limit of thick films, i.e., t D, the domain width follows the law of Kittel, D t. In the limit of thin films, i.e., t D, the domain width increases with D t exp ( ) 1 t In the limit of thick films, the curve follows the prediction of Kittel. The minimum is obtained by decreasing the thickness of the slab when the charges of the two surfaces start to interact. The critical width, i.e., D m =γ m /1.7 Ms 2, is a function of the material and its value is usually between 20 nm and 50 nm. For thinner films the domain width increases rapidly with exponential law: D t exp( 1). The same behaviour is found t analytically [38], after the minimization of the total energy, sum of equations (2.3) and (2.7). In this case the domain width is given by: D(t) = 0.95t exp ( ) πd0 2t (2.8) where D 0 =γ m / 2 πm 2 s is the characteristic dipolar length. Equation (2.8) can be inserted in the expression of the total energy that results [45] : ( ( e t = 2πMs 2 t exp πd )) 0 2t (2.9) The total energy of the single domain state is equal to its magnetostatic energy and it is given by eq. (2.6). In the limit of thick films, i.e., t D, the gain in magnetostatic

5 2.3 Domains separated by walls of finite width 19 energy due to the flux closure is sufficiently high to guarantee that the energy for a stripe domain configuration is lower than for a single domain state. By decreasing the thickness of the film, the gain in magnetostatic energy becomes comparable to the wall energy and the domain formation is less efficient. In the limit of thin films, i.e., t D, the energy for the stripe domain configuration is given by eq. (2.9). From this equation we notice that for every thickness of the slab, the value of the total energy is smaller than for a single domain state. However, the rapid increasing of the domain width, rules out the possibility to have a multi-domain state in a real sample of finite dimensions below a certain critical thickness t c, when the domain width becomes of the same order of the extension of the thin film. 2.3 Domains separated by walls of finite width So far we have considered a system with uniform magnetization within the magnetic domain and with a wall negligible with respect to the domain width. In reality, the Bloch wall between two domains has a finite width and this contributes to the total energy of the system. For certain sets of the parameters of the system, the dimension of the wall becomes comparable with respect to the domain size and the picture of domains separated by narrow walls loses its meaning. In such a case, the magnetization profile can be described by a sine-like function between two maximal values which are a function of the domain width [46]. The goal of this paragraph is to describe films with uniform perpendicular anisotropy taking into account the effect of Bloch walls of finite width between domains of opposite out of the plane magnetization Different contributions of the magnetostatic energy The magnetostatic energy of a sample of volume V is given by: F ms = 2π M(r) H(r) dr (2.10) v The dipolar field H(r) is the result of the interactions of all the magnetic moments distributed in the sample. Therefore the magnetostatic energy density is a non-local quantity, depending on the distribution of all the magnetic charges upon the sample. On the other hand, the anisotropy is a local quantity which depends on the symmetry of the magnetic material considered. In thin films it is possible to write the magnetostatic energy as a sum of an anisotropy-type energy density term, proportional to the area of the film, plus a term describing the influence of the wall fine structure [47]. In this way it is possible to evaluate the rotation of the magnetization which takes place in the region of the wall. With the exception of some particular cases, the integral (2.10) has no analytical solution and the magnetostatic energy can be calculated only numerically. In the case of

6 20 Chapter 2. Stripe domains in thin films a uniformly magnetised film, however, the magnetostatic energy density results: f ms = 2πM 2 s cos 2 θ (2.11) where θ is the angle between the magnetization and the normal of the film. If the film is magnetized parallel to the normal, θ is zero and eq. (2.11) equals eq. (2.6). Since in a system with perpendicular uniaxial anisotropy the angular dependence of the anisotropy energy is of the same form as eq. (2.11), it is convenient to define the following effective anisotropy: k eff = k u 2πM 2 s (2.12) so that the total energy density of the system results: f t = c + k eff sin 2 θ (2.13) In this way the uniformly magnetized system is fully described by anisotropy-type local quantities. It is useful to underline that the magnetostatic energy, although absorbed in the effective anisotropy constant as shown in eq. (2.13), is still a non-local quantity, a function of the mutual interaction of all the spins distributed in the sample. In general the magnetization along the sample is not uniform and the magnetization can rotate forming a Bloch wall, as sketched in Fig. 2.4 for a thin film. Figure 2.4: Infinitely long slab of thickness t and width 2L. The Bloch wall between the two domains has the width δ w =π A k. In case of a thin film not uniformly magnetized along the x direction the magnetostatic energy per unit length can be written in the following way [44]: f ms (L, t) = 2πLt M 2 z (x)dx + Lt2 4 dmz (x) dm z (x ) ln x x dxdx (2.14) dx dx t where L is the length of the unit cell. The first part of the equation corresponds to the anisotropy-like term. It is proportional to the area of the film and it contains the local magnetization M z (x), which is integrated across the wall. The second term represents the correction to the magnetostatic energy due to the rotation of the magnetization, as

7 2.3 Domains separated by walls of finite width 21 shown by the derivative under integration, and it is negative. In the case of uniform magnetization the derivative is zero and eq. (2.14) reduces to eq. (2.11). In the limit of vanishing thickness, i.e., t 0, only the first term remains and the value of the magnetostatic energy can be compared to the one of eq. (2.6). In both cases the demagnetizing factor is equal to one, but the value of the magnetostatic energy is different. In fact, while eq. (2.6) refers to a film uniformly magnetized out of plane, the first term of eq. (2.14) refers to a system with non uniform magnetization. In this latter case both the average value of the magnetic charges and the magnetostatic energy reduce. In general, as will be clear in section 2.3.3, the second term of eq. (2.14) must be considered to guarantee the existence of magnetic domains in systems with uniform uniaxial anisotropy. The goal of section 5.1 will be to show that systems with alternated anisotropy can have magnetic domains even neglecting the correction to the magnetostatic energy Thickness dependence of the domain wall width Before we analyze the formation of a multi-domain state, we study the thickness dependence of a Bloch wall clarifying the role of the different contributions to the magnetostatic energy introduced in the last section. The surface charges of opposite signs on the two sides of a Bloch wall generate a stray field whose intensity is a function of the thickness of the slab. In order to measure the effect of the stray field at first we calculate the energy and the wall profile in the case of vanishing thickness. In this case no stray field is created around the Bloch wall. Therefore the correction to the magnetostatic energy is zero and the system can be described by means of anisotropy-type quantities, as seen in section This state can be obtained by assuming zero magnetostatic energy and the value given by eq. (2.12) for the anisotropy. With reference to Fig. 2.4, we consider a Bloch wall of width δ w dividing an infinite film [48] of thickness t in two regions, oppositely magnetized. In the example, the exchange constant is A=1.05*10 6 erg/cm 3, the uniaxial anisotropy constant k u =1.35*10 7 erg/cm 3, and the spontaneous magnetization is M s =1440 emu/cm 3. The profile of the wall for different thickness is plotted in Fig The shape of the wall is determined by the minimization of the anisotropy energy, the exchange energy and the magnetostatic energy. In particular both the contributions of the magnetostatic energy introduced in section play a role in the determination of the wall structure. In fact, on the one hand the anisotropy-type term tends to enlarge the wall in order to minimize the surface charge density (principle of charges avoiding), on the other hand the correction to the magnetostatic energy tends to narrow it because of the flux closure generated around the wall (see Fig. 2.2). The weight of the two contributions is a function of the thickness of the film as can be seen by defining the following general expression of the effective anisotropy: k eff (t) = k u (t) α(t)2πm 2 s (2.15) The function α(t) measures the contribution of the correction to the magnetostatic energy

8 22 Chapter 2. Stripe domains in thin films and has the following limits: lim α(t) = 1 t 0 lim α(t) = 0 (2.16) t δw The two values are obtained respectively for a film with vanishing thickness and for a thick film. When the thickness of the film tends to zero, the effect of the circular field created around the wall is small and the anisotropy-type term is sufficient to describe the magnetosatic energy. The wall profile obtained in such a case is drawn as a solid line in Fig Figure 2.5: Influence of the thickness t of the film on the width of the Bloch wall. To study the dependency of the magnetostatic energy, the uniaxial anisotropy constant k(t) is considered thickness independent. The dotted and the solid line are the lower and the upper limit for the wall width. The limits correspond to the bulk value (dotted line) and to the zero thickness film (solid line). By increasing the thickness of the film the influence of the circular field created by the surface charges around the Bloch wall increases blocking the expansion of the wall. to In this case the magnetostatic energy tends to 2πM 2 s and the width of the wall tends A δ w = π (2.17) k eff This equation represents the upper limit of the wall width for a system with uniaxial anisotropy k u. The plot in grey in Fig. 2.5 represents the case of an ultrathin film with vanishing thickness, t = 0.3 nm, when the circular magnetic field plays still a negligible role. By increasing the thickness of the film the effect of the field becomes stronger and the wall reduces its extension, as we can see in the dashed profile of Fig In the

9 2.3 Domains separated by walls of finite width 23 limit of thick films, i.e., for t d, the width of the wall is given by equation 1.23, which represents the lower limit for a system with uniaxial anisotropy k u. Hence, by means of the function α(t), the thickness dependence of the circular magnetic field created by the surface charges is described. The validity of this function goes beyond the limit of thin films for which eq. (2.14) has been calculated, but does not describe completely the thickness dependence of the effective anisotropy which are present as well in the uniaxial anisotropy term of eq. (2.15) Transition single domain/multi-domain state In this section, first we want to compare the energy of a single domain state with the one of a multi-domain state, in order to find the critical domain size of the transition for systems with finite walls width. The domain size at the transition is a function of the thickness as well as of the hardness of the material represented by Q. Secondly we show that a multi domain state cannot exist if the dipolar interaction is neglected and only local quantities are considered to describe the system. Finally we compare the transition to the case of systems with negligible wall width. In general, a multi-domain state is energetically favorable in comparison to a single domain state if the energy required to form a wall is lower than the gain in magnetostatic energy due to the stray field. The formation of a multi-domain state can be analyzed in simple terms by considering two domains, opposite magnetized and separated by a Bloch wall, as shown in Fig. 2.4, and comparing the energy obtained with the one of a single domain state. The energy of the wall is independent on the length L of the slab. As a consequence the energy to pay to form the multi-domain state is constant with the length L. On the contrary, the gain in magnetostatic energy density per unit area increases with L. Therefore, for a certain critical value of the length L 0, the energy of the multi-domain state becomes favorable. The situation is well represented by the following equation, obtained by integration of eq. (2.14), using the profile of the Bloch wall M s = M t tanh(x/δ) [41, 44]: F Lt = γ w + µ 0Ms 2 t ln 2c wδ π L (2.18) where c w is a numerical factor. Equation (2.18) gives the energy difference between single and multi-domain state; it is positive for L < L 0 and negative for L>L 0. The term responsible for the domain formation is the logarithm and comes from the non-local part of the magnetostatic energy written in eq. (2.14). This term, which has to be negative in order to reduce the value of the total energy, drives the transition by increasing L, length of the slab. The critical length L 0, normalized to the exchange length l ex, is obtained from eq. (2.18) for F =0: L 0 l ex = 1.9 2Q 1 exp 8.9 2Q 1 t l ex (2.19)

10 24 Chapter 2. Stripe domains in thin films Figure 2.6: Thickness of the film versus L 0, the critical length for the transition to the multi-domain state. The plots are normalized to the exchange length l ex and refer to eq. (2.18) for different values of Q. With increasing Q or decreasing t, l 0 shifts to higher values. In this way the drop in magnetostatic energy balances the increase of the wall energy. The thickness of the film versus the critical length L 0, for different values of Q, is plotted in Fig. (2.6). As already shown in section 2.2 for the case of negligible wall width, the domain size increases rapidly with thickness decreasing, till the size of the sample is reached and the multi domain state cannot exist anymore. This happens because the gain in magnetostatic energy decreases with the thickness and therefore the length L of the slab has to increase to allow domain formation. The dependence of the domain size on the hardness of the material is clear if we consider the effective anisotropy, whose value increases with Q, as can be deduced from eq. (2.15). As a consequence the wall energy increases and, in order to obtain a multi domain configuration, the length L of the slab has to become bigger so that the gain in magnetostatic energy can be comparable to the energy paid to form the wall. Before comparing the different models introduced in the last sections, we show that the existence of the stray field created around the Bloch wall is essential to understand the formation of a multi-domain state. Let us consider once more a system with uniaxial perpendicular effective anisotropy, described only by means of the exchange energy and the anisotropy-type energy terms. If the system is uniformly magnetized out-of-plane, the exchange and the effective anisotropy energy are zero. As soon as a wall is formed and some of the magnetic moments are tilted away from the normal of the film, the effective anisotropy energy and the exchange energy are no more zero. Therefore, if the dipolar

11 2.3 Domains separated by walls of finite width 25 Figure 2.7: Ratio of the total energy between single-domain and multi-domain state vs. the thickness of the slab, normalized to the domain width. If the system is described by means of anisotropy-type terms, the only possible state is the single domain. If the stray field created around the Bloch wall is considered, at a certain critical thickness the system splits into magnetic domains. interaction is not considered in its complete form, the energy of the system increases and the single domain state is the lowest in energy. This shows that the complete non-local dipolar interaction is the fundamental quantity to be considered to explain the formation of a multi-domain state for a system with perpendicular uniaxial anisotropy. In Fig. 2.7 we compare the energies of a multi-domain state with a single-domain state as a function of the thickness, normalized to a fixed value of the domain width. The numerical calculation is performed in the case of an anisotropy-type description (white dots in Fig. 2.7), when the circular magnetic field is not considered, and in the case of total dipolar interaction (black dots in Fig. 2.7). If the correction to the dipolar energy is not considered, the ratio of the total energy is constant with the thickness and the single domain state has lower energy than the multi-domain state. This happens because the magnetostatic energy density is thickness independent for both states. If the correction is considered, the magnetostatic energy density is no more thickness independent. As a consequence the total energy of the multi-domain state decreases linearly with the thickness allowing the transition to the multi-domain state. The analysis carried out in this section has the purpose to determine the conditions for which it is favorable to have a multi-domain configuration. To complete the picture, we want to study the influence of the presence of the domain wall in the determination of the domain size by comparing eq. (2.18) with eq. (2.8), calculated with the assumption of uniform magnetization and negligible wall width. The two curves are plotted in Fig. 2.8,

12 26 Chapter 2. Stripe domains in thin films Figure 2.8: L 0, critical length for the transition to the multi-domain state versus t, thickness of the film. The plots shown, normalized to the exchange length l ex, are for a value of Q=1.3. The black plot refers to eq. (2.8), where the wall-width is neglected. The dashed plot refers to eq. (2.18), where the wall width is considered finite. The black points result from numeric simulation performed with periodic boundary conditions. for Q=1.3. Fixing an arbitrary value of the thickness of the slab we notice that the domain size is significantly larger if the wall is included in the description. In this case the average perpendicular magnetization around the region of the wall is smaller than for a system with negligible wall width. As a consequence the gain in magnetostatic energy due to the flux closure is limited by the presence of the wall and the transition to the multi-domain state shifts to larger values of the domain size. Therefore, the assumptions of uniform magnetization and negligible wall width made in section 2.2, cause an overestimation of the gain in magnetostatic energy and a significant underestimation of the value of the domain size. To calculate correctly the size of the unit cell in the case of finite wall width, we have to bound the system periodically. In this way a new Bloch wall forms at the edge of the slab and the unit cell of the multi-domain state is given by the sum of the domain width and of the wall width. This choice, not considered in the derivation of eq. (2.18) [41], yields a further shift of the value of the domain size, as shown in Fig The difference is due to the presence of the Bloch wall at the edge of the slab and to the consequent reduction of the average magnetization around it.

### Spatially varying magnetic anisotropies in ultrathin films

Spatially varying magnetic anisotropies in ultrathin films Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt der Mathematisch-Naturwissenschaftlich-Technischen

### Advanced Lab Course. MOKE Microscopy

Advanced Lab Course MOKE Microscopy M209 Stand: 2016-04-08 Aim: Characterization of magnetization reversal of magnetic bulk and thin film samples by means of magneto-optical Kerr effect and recording of

### CBE 6333, R. Levicky 1. Potential Flow

CBE 6333, R. Levicky Part I. Theoretical Background. Potential Flow Potential Flow. Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### Reflection and Refraction

Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

### Chapter 3. Gauss s Law

3 3 3-0 Chapter 3 Gauss s Law 3.1 Electric Flux... 3-2 3.2 Gauss s Law (see also Gauss s Law Simulation in Section 3.10)... 3-4 Example 3.1: Infinitely Long Rod of Uniform Charge Density... 3-9 Example

### Chapter 22: Electric Flux and Gauss s Law

22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

### Ajit Kumar Patra (Autor) Crystal structure, anisotropy and spin reorientation transition of highly coercive, epitaxial Pr-Co films

Ajit Kumar Patra (Autor) Crystal structure, anisotropy and spin reorientation transition of highly coercive, epitaxial Pr-Co films https://cuvillier.de/de/shop/publications/1306 Copyright: Cuvillier Verlag,

### Magnetic Field of a Circular Coil Lab 12

HB 11-26-07 Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment- coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator,

### Gasses at High Temperatures

Gasses at High Temperatures The Maxwell Speed Distribution for Relativistic Speeds Marcel Haas, 333 Summary In this article we consider the Maxwell Speed Distribution (MSD) for gasses at very high temperatures.

### Analytical and micromagnetic study of a Néel domain wall

PHYSICAL REVIEW B 77, 449 008 Analytical and micromagnetic study of a Néel domain wall K. Rivkin, K. Romanov, Ar. Abanov, Y. Adamov, and W. M. Saslow Department of Physics, Texas A &M University, College

### Blasius solution. Chapter 19. 19.1 Boundary layer over a semi-infinite flat plate

Chapter 19 Blasius solution 191 Boundary layer over a semi-infinite flat plate Let us consider a uniform and stationary flow impinging tangentially upon a vertical flat plate of semi-infinite length Fig

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### CHAPTER 24 GAUSS S LAW

CHAPTER 4 GAUSS S LAW 4. The net charge shown in Fig. 4-40 is Q. Identify each of the charges A, B, C shown. A B C FIGURE 4-40 4. From the direction of the lines of force (away from positive and toward

Last Name: First Name: Physics 102 Spring 2006: Exam #2 Multiple-Choice Questions 1. A charged particle, q, is moving with speed v perpendicular to a uniform magnetic field. A second identical charged

### Exam 1 Practice Problems Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8 Spring 13 Exam 1 Practice Problems Solutions Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical

### Physics 2212 GH Quiz #4 Solutions Spring 2015

Physics 1 GH Quiz #4 Solutions Spring 15 Fundamental Charge e = 1.6 1 19 C Mass of an Electron m e = 9.19 1 31 kg Coulomb constant K = 8.988 1 9 N m /C Vacuum Permittivity ϵ = 8.854 1 1 C /N m Earth s

### Oscillations. Vern Lindberg. June 10, 2010

Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1

### Home Work 9. i 2 a 2. a 2 4 a 2 2

Home Work 9 9-1 A square loop of wire of edge length a carries current i. Show that, at the center of the loop, the of the magnetic field produced by the current is 0i B a The center of a square is a distance

### Module 7. Transformer. Version 2 EE IIT, Kharagpur

Module 7 Transformer Version EE IIT, Kharagpur Lesson 4 Practical Transformer Version EE IIT, Kharagpur Contents 4 Practical Transformer 4 4. Goals of the lesson. 4 4. Practical transformer. 4 4.. Core

### A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle

A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle Page 1 of 15 Abstract: The wireless power transfer link between two coils is determined by the properties of the

### Analysis of Stresses and Strains

Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we

### Lecture 8. Generating a non-uniform probability distribution

Discrete outcomes Lecture 8 Generating a non-uniform probability distribution Last week we discussed generating a non-uniform probability distribution for the case of finite discrete outcomes. An algorithm

### Gauss's Law. Gauss's Law in 3, 2, and 1 Dimension

[ Assignment View ] [ Eðlisfræði 2, vor 2007 22. Gauss' Law Assignment is due at 2:00am on Wednesday, January 31, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed.

### There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all).

Data acquisition There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all). Some deal with the detection system.

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### PSS 27.2 The Electric Field of a Continuous Distribution of Charge

Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

### Fall 12 PHY 122 Homework Solutions #8

Fall 12 PHY 122 Homework Solutions #8 Chapter 27 Problem 22 An electron moves with velocity v= (7.0i - 6.0j)10 4 m/s in a magnetic field B= (-0.80i + 0.60j)T. Determine the magnitude and direction of the

### 3. Diffusion of an Instantaneous Point Source

3. Diffusion of an Instantaneous Point Source The equation of conservation of mass is also known as the transport equation, because it describes the transport of scalar species in a fluid systems. In this

### Problem 4.48 Solution:

Problem 4.48 With reference to Fig. 4-19, find E 1 if E 2 = ˆx3 ŷ2+ẑ2 (V/m), ε 1 = 2ε 0, ε 2 = 18ε 0, and the boundary has a surface charge density ρ s = 3.54 10 11 (C/m 2 ). What angle does E 2 make with

### Graphical Presentation of Data

Graphical Presentation of Data Guidelines for Making Graphs Titles should tell the reader exactly what is graphed Remove stray lines, legends, points, and any other unintended additions by the computer

### Torsion Tests. Subjects of interest

Chapter 10 Torsion Tests Subjects of interest Introduction/Objectives Mechanical properties in torsion Torsional stresses for large plastic strains Type of torsion failures Torsion test vs.tension test

### Electric field outside a parallel plate capacitor

Electric field outside a parallel plate capacitor G. W. Parker a) Department of Physics, North Carolina State University, Raleigh, North Carolina 7695-80 Received 7 September 00; accepted 3 January 00

### IB Mathematics SL :: Checklist. Topic 1 Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications.

Topic 1 Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications. Check Topic 1 Book Arithmetic sequences and series; Sum of finite arithmetic series; Geometric

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### Announcements. Moment of a Force

Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2-D and 3-D cases Moment of

### 1 Measuring resistive devices

1 Measuring resistive devices 1.1 Sourcing Voltage A current can only be maintained in a closed circuit by a source of electrical energy. The simplest way to generate a current in a circuit is to use a

### Electromagnetism Laws and Equations

Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2

### Physics 1120: Simple Harmonic Motion Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

### * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES

ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES The purpose of this lab session is to experimentally investigate the relation between electric field lines of force and equipotential surfaces in two dimensions.

### Physics 2220 Module 09 Homework

Physics 2220 Module 09 Homework 01. A potential difference of 0.050 V is developed across the 10-cm-long wire of the figure as it moves though a magnetic field perpendicular to the page. What are the strength

### Handout 7: Magnetic force. Magnetic force on moving charge

1 Handout 7: Magnetic force Magnetic force on moving charge A particle of charge q moving at velocity v in magnetic field B experiences a magnetic force F = qv B. The direction of the magnetic force is

### Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions

Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem

### 0.1 Dielectric Slab Waveguide

0.1 Dielectric Slab Waveguide At high frequencies (especially optical frequencies) the loss associated with the induced current in the metal walls is too high. A transmission line filled with dielectric

### Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II Welcome to the second lesson of the fifth module which is on Bending of Beams part

### Two-Dimensional Conduction: Shape Factors and Dimensionless Conduction Heat Rates

Two-Dimensional Conduction: Shape Factors and Dimensionless Conduction Heat Rates Chapter 4 Sections 4.1 and 4.3 make use of commercial FEA program to look at this. D Conduction- General Considerations

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 29a. Electromagnetic Induction Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the

### Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

### D Alembert s principle and applications

Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Stress Analysis Verification Manual

Settle3D 3D settlement for foundations Stress Analysis Verification Manual 007-01 Rocscience Inc. Table of Contents Settle3D Stress Verification Problems 1 Vertical Stresses underneath Rectangular Footings

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### Rotational Motion. Symbol Units Symbol Units Position x (m) θ (rad) (m/s) " = d# Source of Parameter Symbol Units Parameter Symbol Units

Introduction Rotational Motion There are many similarities between straight-line motion (translation) in one dimension and angular motion (rotation) of a rigid object that is spinning around some rotation

### 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration

Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### Toy Blocks and Rotational Physics

Toy Blocks and Rotational Physics Gabriele U. Varieschi and Isabel R. Jully, Loyola Marymount University, Los Angeles, CA Have you ever observed a child playing with toy blocks? A favorite game is to build

### Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

### Electromagnetic Induction

Electromagnetic Induction "Concepts without factual content are empty; sense data without concepts are blind... The understanding cannot see. The senses cannot think. By their union only can knowledge

### p atmospheric Statics : Pressure Hydrostatic Pressure: linear change in pressure with depth Measure depth, h, from free surface Pressure Head p gh

IVE1400: n Introduction to Fluid Mechanics Statics : Pressure : Statics r P Sleigh: P..Sleigh@leeds.ac.uk r J Noakes:.J.Noakes@leeds.ac.uk January 008 Module web site: www.efm.leeds.ac.uk/ive/fluidslevel1

### Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements)

Test - A2 Physics Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Time allocation 40 minutes These questions were ALL taken from the June 2010

### Global MRI and Dynamo in Accretion Discs

Global MRI and Dynamo in Accretion Discs Evy Kersalé PPARC Postdoctoral Research Associate Dept. of Appl. Maths University of Leeds Collaboration: D. Hughes & S. Tobias (Appl. Maths, Leeds) N. Weiss &

### Transmission through the quadrupole mass spectrometer mass filter: The effect of aperture and harmonics

Transmission through the quadrupole mass spectrometer mass filter: The effect of aperture and harmonics A. C. C. Voo, R. Ng, a) J. J. Tunstall, b) and S. Taylor Department of Electrical and Electronic

### Basic Principles of Magnetic Resonance

Basic Principles of Magnetic Resonance Contents: Jorge Jovicich jovicich@mit.edu I) Historical Background II) An MR experiment - Overview - Can we scan the subject? - The subject goes into the magnet -

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

### Dynamic Domains in Strongly Driven Ferromagnetic Films

Dynamic Domains in Strongly Driven Ferromagnetic Films A dissertation accepted by the Department of Physics of Darmstadt University of Technology for the degree of Doktor der Naturwissenschaften (Dr. rer.

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### > 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### Maths for Computer Graphics

Analytic Geometry Review of geometry Euclid laid the foundations of geometry that have been taught in schools for centuries. In the last century, mathematicians such as Bernhard Riemann (1809 1900) and

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### Projectile Motion. Equations of Motion for Constant Acceleration

Projectile Motion Projectile motion is a special case of two-dimensional motion. A particle moving in a vertical plane with an initial velocity and experiencing a free-fall (downward) acceleration, displays

### Physics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd

Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle

### Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

### Thermal Diffusivity, Specific Heat, and Thermal Conductivity of Aluminum Oxide and Pyroceram 9606

Report on the Thermal Diffusivity, Specific Heat, and Thermal Conductivity of Aluminum Oxide and Pyroceram 9606 This report presents the results of phenol diffusivity, specific heat and calculated thermal

### Physics 9 Fall 2009 Homework 8 - Solutions

1. Chapter 34 - Exercise 9. Physics 9 Fall 2009 Homework 8 - s The current in the solenoid in the figure is increasing. The solenoid is surrounded by a conducting loop. Is there a current in the loop?

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Rotation: Kinematics

Rotation: Kinematics Rotation refers to the turning of an object about a fixed axis and is very commonly encountered in day to day life. The motion of a fan, the turning of a door knob and the opening

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### Solution Derivations for Capa #10

Solution Derivations for Capa #10 1) A 1000-turn loop (radius = 0.038 m) of wire is connected to a (25 Ω) resistor as shown in the figure. A magnetic field is directed perpendicular to the plane of the

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### CONSTANT ELECTRIC CURRENT AND THE DISTRIBUTION OF SURFACE CHARGES 1

CONSTANT ELECTRIC CURRENT AND THE DISTRIBUTION OF SURFACE CHARGES 1 Hermann Härtel Guest scientist at Institute for Theoretical Physics and Astrophysics University Kiel ABSTRACT Surface charges are present,

### Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P

Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P r + v r A. points in the same direction as v. B. points from point

### 1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius?

CHAPTER 3 SOURCES O THE MAGNETC ELD 1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 8 µ T at the loop center. What is the loop radius? Equation 3-3, with

### Yield Criteria for Ductile Materials and Fracture Mechanics of Brittle Materials. τ xy 2σ y. σ x 3. τ yz 2σ z 3. ) 2 + ( σ 3. σ 3

Yield Criteria for Ductile Materials and Fracture Mechanics of Brittle Materials Brittle materials are materials that display Hookean behavior (linear relationship between stress and strain) and which

### Orbits of the Lennard-Jones Potential

Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Chapter 7 Direct-Current Circuits

Chapter 7 Direct-Current Circuits 7. Introduction...7-7. Electromotive Force...7-3 7.3 Resistors in Series and in Parallel...7-5 7.4 Kirchhoff s Circuit Rules...7-7 7.5 Voltage-Current Measurements...7-9

### 2 A Dielectric Sphere in a Uniform Electric Field

Dielectric Problems and Electric Susceptability Lecture 10 1 A Dielectric Filled Parallel Plate Capacitor Suppose an infinite, parallel plate capacitor with a dielectric of dielectric constant ǫ is inserted