u B = B 2 2 . u_B = \frac {B^2} {2\mu} u. . A magnetic field is a region in space where a moving charge or permanent magnet feels a force. We may therefore write I = B/ ( 0 n), and U = ( 0 n 2 A)* (B/ ( 0 n)) 2 = (B 2 / (2 0 )) (A*). the energy density is altered. Kinetic by OpenStax offers access to innovative study tools designed to help you maximize your learning potential. 0000000816 00000 n Similarly, an inductor has the capability to store energy, but in its magnetic field. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. To understand where this formula comes from, lets consider the long, cylindrical solenoid of the previous section. 0000001596 00000 n 0000008242 00000 n Magnetic Field of a Toroidal Solenoid We know that (947) where is the number of turns per unit length of the solenoid, the radius, and the length. are not subject to the Creative Commons license and may not be reproduced without the prior and express written Also, the energy storing capacity of the magnetic field is greater than the . Distance between two plates = d Hence, electric field intensity,E = V/X= V/d A positively charged particle,P experience an electric force F = q.E F = q. The potential energy on one dipole from the magnetic field from the other is: . A magnetic field is produced by moving electric charges and intrinsic magnetic moments of elementary particles associated with a fundamental quantum property known as spin. It is equal to the amount of current required to generate current through the inductor if energy is stored in a . Based on this magnetic field, we can use Equation 14.22 to calculate the energy density of the magnetic field. At any instant, the magnitude of the induced emf is \(\epsilon = Ldi/dt\), where i is the induced current at that instance. This argument also holds when \(r < R_1\); that is, in the region within the inner cylinder. The energy density of an electromagnetic wave can be calculated with help of the formula of energy density which is U = \[\frac{1}{2} \epsilon _oE^2 + \frac{1}{2\mu _0} B^2\]. trailer Similarly, an inductor has the capability to store energy, but in its magnetic field. A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents,: ch1 and magnetic materials. then you must include on every digital page view the following attribution: Use the information below to generate a citation. With the substitution of Equation 14.3.12, this becomes U = 1 2LI2. To understand where this formula comes from, lets consider the long, cylindrical solenoid of the previous section. I know KE = 1/2mv^2 Using KE = 1/2mv^2 and saying KE = 5.8 x 10^-17, and m = 9.10938 x 10^-31 KG I get that v= 11284559 m/s each coil is connected to its own battery. Magnetization can be expressed in terms of magnetic intensity as. [/latex], [latex]U={\int }_{{R}_{1}}^{{R}_{2}}dU={\int }_{{R}_{1}}^{{R}_{2}}\frac{{\mu }_{0}{I}^{2}}{8{\pi }^{2}{r}^{2}}\left(2\pi rl\right)dr=\frac{{\mu }_{0}{I}^{2}l}{4\pi }\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{R}_{2}}{{R}_{1}},[/latex], [latex]\frac{L}{l}=\frac{{\mu }_{0}}{2\pi }\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{R}_{2}}{{R}_{1}}. Energy Density Formula. Formula where, 0 denotes permeability of free space constant, I denotes the magnitude of electric current r denotes the distance in meters This power is expressed in terms of the Poynting vector. Almost 100% orientation is observed in blood samples exposed to a static field of 4 T. Interestingly, neither the direction nor the degree . How much energy is stored in the inductor of Example 14.3.1 after the current reaches its maximum value? endstream endobj 62 0 obj<> endobj 63 0 obj<> endobj 64 0 obj<> endobj 65 0 obj<> endobj 66 0 obj<>stream ThereforeInduced EMF = (change in Magnetic Flux Density x Area)/change in Time. Figure \(\PageIndex{1}\) shows two long, concentric cylindrical shells of radii \(R_1\) and \(R_2\). It is more common, however, to define it by the Lorentz-force equation. Magnetic fields affect the alignment of electrons in an atom, and can cause physical force to develop between atoms across space just as with electric fields developing force between electrically charged particles. The associated circuit equation is The electric energy input into the ideal coil due to the flow of current i in time dt is Assuming for the time being that the armature is held fixed at position x, all the input energy is stored in the magnetic field. x- [ 0}y)7ta>jT7@t`q2&6ZL?_yxg)zLU*uSkSeO4?c. R -25 S>Vd`rn~Y&+`;A4 A9 =-tl`;~p Gp| [`L` "AYA+Cb(R, *T2B- Flux density dependency on the nature of the magnetic coupling material of VEH magnet . stored in the surrounding magnetic field. The potential energy of a magnet or magnetic moment in a magnetic field is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to: 0000014976 00000 n the volume of the magnetic field is modified. In this limit, there is no coaxial cable. Since the energy density of the magnetic field is \[u_m = \dfrac{B^2}{2\mu_0}\nonumber\] the energy stored in a cylindrical shell of inner radius, From Equation \ref{14.22}, \[U = \dfrac{1}{2}LI^2,\] where. Physics - E&M: Inductance (8 of 20) Energy Stored in a Magnetic Field 39,620 views Dec 7, 2014 455 Dislike Michel van Biezen 879K subscribers Visit http://ilectureonline.com for more math and. Answer (1 of 6): when ever a particle or a object come in the influence of some field or particle then it interacts with the field or particle. The total energy of the magnetic field is given by the sum of the energy density of the single points. However Feynman writes in Section 27-4 of his well known course: Substituting in equation (4) B = 0 (H + H) B = 0 (1 + ) H. The quantity (1 + ) is called relative magnetic permeability and is denoted by r. It is a dimensionless quantity. 51 26 HUMoGQwQMaAR9V"V_E! 0000001220 00000 n If you are redistributing all or part of this book in a print format, The formula is the sum of the energy density of electric and magnetic fields. Based on this magnetic field, we can use Equation \ref{14.22} to calculate the energy density of the magnetic field. It should be noted that the total stored energy in the magnetic field depends upon the final or steady-state value of the current and is independent of the manner in which the current has increase or time it has taken to grow. Magnetic energy and electrostatic potential energy are related by Maxwell's equations. Therefore, the power absorbed by the inductor is. 0000002201 00000 n B =BA = BAcos For a varying magnetic field the magnetic flux is dB through an infinitesimal area dA: dB = BdA The surface integral gives the total magnetic flux through the surface. The magnetic field of a solenoid near the ends approaches half of the magnetic field at the center, that is the magnetic field gradually decreases from the center to the ends. In the case of a magnetic field or an inductor, the energy density is given by, U=1B 2 /2 0. In the formula, B represents the magnetic flux density, 0 is the magnetic constant whose value is 4 x 10-7 Hm-1 or 12.57 x 10-7 Hm-1, N represents the number of turns, and I is the current flowing through the solenoid. mMVY+LQsEPaBZ\X~0Z[pdV!ZXu>%19kNbPb]wZwta+um q@ @I"/Z8orP?fn{ O!uln u0:ZjH ; ]GO/tx\T( explicit formula for the energy stored in a magnetic field. In the case of electrical energy. 0000002442 00000 n The field of electricity and magnetism is also used to store energy. Below are the online magnetic field strength calculators to find around a wire, magnetic field strength inside a loop and magnetic field inside a solenoid. for example if a charge come near another charge then it feels electrostatic potential similarly if a charge particle is placed in magnetic field it inte. The above equation also tells us that the magnetic field is uniform over the cross-section of the solenoid. 1. 0000015215 00000 n This is known as permeability of free space and has a = / A). This page titled 14.4: Energy in a Magnetic Field is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. Legal. of circuits (labeled to ), each carrying a current . %PDF-1.4 % Magnetic Force Acting on a Moving Charge in the Presence of Magnetic Field A change 'a' is moving with a velocity 'v' making an angle '' with the field direction. lb9N(r}`}QpoRHrVVV%q *ia1Ejijs0 Now (a) determine the magnetic energy stored per unit length of the coaxial cable and (b) use this result to find the self-inductance per unit length of the cable. Based on this magnetic field, we can use Equation 14.22 to calculate the energy density of the magnetic field. Let us now obtain an explicit formula for the energy stored in a magnetic field. xb```V yAb,xOvhG|#T]IDWwVeK]jYG|lI Particle in a Magnetic Field. An electron has a kinetic energy of 5.80 10-17 J. To increase the inductance, we could either increase the outer radius (\(R_2\)) or decrease the inner radius (\(R_1\)). 8$5z2vC@z)}7|d\\7S&1g)vBJf.^[*24?Y3]=~pFgEka[Z\}DJL/d4Ckj : ch13 : 278 A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. 0000024211 00000 n U E = E 2 /2. 0 Consider, again, our circuit with two coils wound on top of one another. The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. For this reason the energy of a magnetic field shifts while: 1.) The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. I is the current intensity, in Ampere. Find the value of the magnetic field inside a solenoid of 5 m and 500 turns per unit length if 10A of current is passing through it. The total energy stored per volume is the energy density of the electromagnetic wave (U), which is the sum of electric field energy density (U E) and magnetic field energy density (U B ). 0000024440 00000 n To do that, we have to describe how much energy there is in any volume element of space, and also the rate of energy flow. wG xR^[ochg`>b$*~ :Eb~,m,-,Y*6X[F=3Y~d tizf6~`{v.Ng#{}}jc1X6fm;'_9 r:8q:O:8uJqnv=MmR 4 The . All the magnetic energy of the cable is therefore stored between the two conductors. As a result, the energy density of . Energy is required to establish a magnetic field. Since we know that the NMR frequency is directly proportional to the magnetic strength, we calculate the magnetic field at 400 MHz: B 0 = (400 MHz/60MHz) x 1.41 T = 9.40 T Look under applications. According to David C Jiles, magnetic field intensity definition is as follows: " A magnetic field intensity or strength of 1 ampere per meter is produced at the center of a single circular coil of conductor of diameter 1 meter when it carries a current of 1 ampere.". Show: which is used to calculate the energy stored in an inductor. Consider the two circuits sharing a common return plane shown in Fig. Example 1. are licensed under a, Heat Transfer, Specific Heat, and Calorimetry, Heat Capacity and Equipartition of Energy, Statements of the Second Law of Thermodynamics, Conductors, Insulators, and Charging by Induction, Calculating Electric Fields of Charge Distributions, Electric Potential and Potential Difference, Motion of a Charged Particle in a Magnetic Field, Magnetic Force on a Current-Carrying Conductor, Applications of Magnetic Forces and Fields, Magnetic Field Due to a Thin Straight Wire, Magnetic Force between Two Parallel Currents, Applications of Electromagnetic Induction, Maxwells Equations and Electromagnetic Waves, (a) A coaxial cable is represented here by two hollow, concentric cylindrical conductors along which electric current flows in opposite directions. The total energy stored in the magnetic field when the current increases from 0 to I in a time interval from 0 to t can be determined by integrating this expression: \[U = \int_0^t Pdt' = \int_0^t L\dfrac{di}{dt'}idt' = L\int_0^l idi = \dfrac{1}{2}LI^2. The Earth's magnetic field is also important for navigation, as it is used by compasses to find magnetic north. nQt}MA0alSx k&^>0|>_',G! The capacitance per unit length of the cable has already been calculated. Calculating the induced EMF. Want to cite, share, or modify this book? We can see this by considering an arbitrary inductor through which a changing current is passing. Because of the cylindrical symmetry, \(\vec{B}\) is constant along the path, and \[\oint \vec{B} \cdot d\vec{l} = B(2\pi r) = \mu_0 I.\] This gives us \[B = \dfrac{\mu_0I}{2\pi r}. Characteristics: 0000002739 00000 n The circuit equations are thus, We intimated previously that the energy stored in an inductor is actually A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. d\stackrel{\to }{\textbf{l}}=B\left(2\pi r\right)={\mu }_{0}I. startxref The magnetic field both inside and outside the coaxial cable is determined by Ampre's law. (V/d) Or,. Fields have two measures: a field force and a field flux. M>G`oG;ENEQo!!bKk^Q=\ $ Approximate Cyg A (Figure 5.12 ) by two spherical lobes of radius R 30 kpc and luminosity L / 2 each, where L is the total luminosity of Cyg A: I$9z/ QbJ 3/D^9u*/UP!lRA;4i}Y7W 9 0000004190 00000 n Maxwell's first equation says the tendency of the elctric field to spread out (or contract) at any point is proportional to the electric charge at that point. a = q.V/m.d By the third eqn of motion v = u + 2as Putting the values v = O +2 (q.V/m.d)d v = 2q.V/m mv = 2q.V The equation is written. Although derived for a special case, this equation gives the energy stored in the magnetic field of any inductor. Therefore, Induced EMF = (Br2n)/t. Magnetic field magnitude = B = Derivation of the Formula B = refers to the magnetic field magnitude in Tesla (T) = refers to the permeability of free space () The magnetic field is a field, produced by electric charges in motion. For example, if the coil bobbin width is 30mm, a distance of 15mm is at the coil edge. As an Amazon Associate we earn from qualifying purchases. The magnetic induction, B, can be defined in a manner similar to E as proportional to the force per unit pole strength when a test magnetic pole is brought close to a source of magnetization. For electromagnetic waves, both the electric and magnetic fields play a role in the transport of energy. Magnetic field lines represent the direction in which a magnetic north pole would move in the field. In most labs this magnetic field is somewhere between 1 and 21T. So, as per conservation of the magnetic flux Law. Using the formula for magnetic field we have, B = o IN/L = 4 10 -7 (400/2) 5 = 4 10 -7 200 5 = 12.56 10 -4 T Problem 2. The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. Key Points. 2y.-;!KZ ^i"L0- @8(r;q7Ly&Qq4j|9 Besides, the unit of a magnetic field is Tesla (T). Formula of the Magnetic Field in Solenoi d To apply Ampere's law, consider an imaginary amperian loop in the shape of a rectangle \ (abcd,\) as shown in the below figure. n3kGz=[==B0FX'+tG,}/Hh8mW2p[AiAN#8$X?AKHI{!7. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. KEPP GENSET is the first commercial-ready magnetic-drive power generator, using the U.S. Patented torque amplifier methodology. (A*) is the volume surrounded by the coil. 0000015417 00000 n Magnetic Resonance in Chemistry and Medicine. 27-2 Energy conservation and electromagnetism. The energy stored in a magnetic field is equal to the work needed to produce a current through the inductor. Q T;GPzu. Jn0~6H J%%HIaYeB(M2{.~Xm$Vdvbd?8?P50Ft8O"[2&zQbu&gTYGKw_@Or(q0J&8sn[JR@ed1%:8M ,-q, FlL95XENE-AF& m; 76 0 obj<>stream Based on this magnetic field, we can use Equation 11.3.5 to calculate the energy density of the magnetic field. The energy stored in the solenoid when a current flows through it Simply put, magnetic energy is the energy that operates within a magnetic field. N')].uJr (900)]. Equation (1) can be written as. It moves on a circular path that is perpendicular to a uniform magnetic field of magnitude 5.10 10-5 T. Determine the radius of the path? The magnetic field both inside and outside the coaxial cable is determined by Ampres law. RBCs in a strong static magnetic field tend to orient themselves with the disk plane along the field, a result of the anisotropy of the cell's diamagnetic response. We recommend using a Now this flux is of two types, (a) r this is remanent flux of the magnet and (b) d this is demagnetizing flux. The expression for magnetic potential energy can be developed from the expression for the magnetic torque on a current loop. Experimentally, we found that a magnetic force acts on the moving charge and is given by F B = q ( V B ). Strategy The magnetic field both inside and outside the coaxial cable is determined by Ampre's law. A magnetic field is invisible to the naked eye, but that does mean that the effects of magnetic energy are not felt. A. Bifone, in Encyclopedia of Condensed Matter Physics, 2005 RBCs in Static Magnetic Fields. Magnetic forces exist between the poles of magnets; like poles repel and unlike poles attract. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. Energy density is defined as the amount of energy accumulated in a system per unit volume. Magnetic Field Energy Density -- from Eric Weisstein's World of Physics In cgs, the energy density contained in a magnetic field B is U = {1\over 8\pi} B^2, and in MKS is given by U = {1\over 2\mu_0} B^2, where \mu_0 is the permeability of free space. The ampere per square meter is the unit of magnetic field strength. The self-inductance per unit length is determined based on this result and Equation \ref{14.22}. Magnetic field coupling (also called inductive coupling) occurs when energy is coupled from one circuit to another through a magnetic field. citation tool such as, Authors: Samuel J. Ling, William Moebs, Jeff Sanny. Suppose that Answer: The magnitude of the magnetic field can be calculated using the formula: The magnitude of the magnetic field is 6.00 x 10 -6 T, which can also be written as (micro-Tesla). We want now to write quantitatively the conservation of energy for electromagnetism. Then we can write that = B.A, where B is the flux density. Magnetic field strength is a physical number that is one of the most fundamental measurements of the magnetic field's intensity. U = um(V) = (0nI)2 20 (Al) = 1 2(0n2Al)I2. 0000004664 00000 n All magnets must have a north and south pole. [/latex], [latex]{u}_{\text{m}}=\frac{{B}^{2}}{2{\mu }_{0}}=\frac{{\mu }_{0}{I}^{2}}{8{\pi }^{2}{r}^{2}},[/latex], [latex]{u}_{\text{m}}=\frac{{B}^{2}}{2{\mu }_{0}}=\frac{{\mu }_{0}{I}^{2}}{8{\pi }^{2}{r}^{2}}. consent of Rice University. Faraday's law states: Induced EMF is equal to the rate of change of magnetic flux. 0000004312 00000 n solenoid. The total energy stored in the magnetic field when the current increases from 0 to I in a time interval from 0 to t can be determined by integrating this expression: Check Your Understanding How much energy is stored in the inductor of Example 14.2 after the current reaches its maximum value? Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. The magnetic field strength B min that minimizes the total energy in the relativistic particles and magnetic fields implied by the luminous synchrotron source can be estimated with Equation 5.109. The energy stored in any part of the electromagnetic wave is the sum of electric field energy and magnetic field energy. The energy stored in the solenoid when a current flows through it is (946) where is the self-inductance. The formula for the energy stored in a magnetic field is E = 1/2 LI 2. The intensity B of the magnetic field of a solenoid composed of coils wound in air (that is, without a ferromagnetic core) can be calculated using the following formula: where: 0 = 4 x 10 -7 H/m is the magnetic constant (vacuum permeability) N is the number of turns. endstream endobj 67 0 obj<> endobj 68 0 obj<> endobj 69 0 obj<>stream The energy stored in the magnetic field of an inductor can be written as: w = 1 2Li2 (2) w L. Where w is the stored energy in joules, L is the inductance in Henrys, and i is the current in amperes. The field force is the amount of "push" that a field exerts over a certain distance. "F$H:R!zFQd?r9\A&GrQhE]a4zBgE#H *B=0HIpp0MxJ$D1D, VKYdE"EI2EBGt4MzNr!YK ?%_&#(0J:EAiQ(()WT6U@P+!~mDe!hh/']B/?a0nhF!X8kc&5S6lIa2cKMA!E#dV(kel }}Cq9 Consider an ideal solenoid. [/latex], [latex]B=\frac{{\mu }_{0}I}{2\pi r}. Consider a system Corresponding the stored energy is. The electric and magnetic fields can be written in terms of a scalar and a vector potential: B = A, E = . The Ampere's law is reproduced as follows: Again using the infinite solenoid approximation, we can assume that the magnetic field is essentially constant and given by \(B = \mu_0 nI\) everywhere inside the solenoid. The 3D coordinate of a magnetic dipole pair can be seen in Fig. The Magnetic Field Equation can then be described by Ampere's law and is solely governed by the conduction current. vt`30@,QbclppAw4u0vX> c`:r86b` ~` i Enter zero for the magnetic at the center of the coil/solenoid. New York: Springer-Verlag, 1986. Creative Commons Attribution License 0000002167 00000 n U=1 0 E 2 /2. Magnetic field in a solenoid formula is given as B = 0 nl. Total flux flowing through the magnet cross-sectional area A is . The difference in energy between aligned and anti-aligned is. After the integration is carried out, we have a closed-form solution for part (a). 1. Find the maximum energy stored by an inductor with an inductance of 5.0 H and a resistance of 2.0 V when the inductor is connected to a . Energy density can be written as \text {u}_\text {B} = \frac {\text {B}^2} {2\mu} uB = 2B2 . Our mission is to improve educational access and learning for everyone. Nuclear Magnetic Resonance. 0000001983 00000 n Energy density can be written as. (c) The cylindrical shell is used to find the magnetic energy stored in a length, https://openstax.org/books/university-physics-volume-2/pages/1-introduction, https://openstax.org/books/university-physics-volume-2/pages/14-3-energy-in-a-magnetic-field, Creative Commons Attribution 4.0 International License, Explain how energy can be stored in a magnetic field, Derive the equation for energy stored in a coaxial cable given the magnetic energy density, We determine the magnetic field between the conductors by applying Ampres law to the dashed circular path shown in. The formula for the energy stored in a magnetic field is E = 1/2 LI 2. \label{14.19}\], With the substitution of Equation 14.3.12, this becomes, Although derived for a special case, this equation gives the energy stored in the magnetic field of any inductor. Jun 29, 2022 OpenStax. The total energy stored in the magnetostatic field is obtained by integrating the energy density, W B, over all space (the element of volume is d ): Field Force and Field Flux It also. mGsPsW#UKIpGR Magnetic flux = Magnetic field strength x Area = BA. Thus where dW f is the change in field energy in time dt. The magnetic flux through the th circuit is written [cf., Eq. The direction of the magnetic field can be determined using the "right hand rule", by pointing the thumb of your right hand in the direction of the current. The formula for energy density of electromagnetic field in electrodynamics is $$\frac{1}{8\pi} (\vec E\cdot\vec D+\vec B\cdot\vec H).$$ This formula appears in all general physics courses I looked at. Note 7: Enter the core relative permeability constant, k. endstream endobj 52 0 obj<> endobj 53 0 obj<> endobj 54 0 obj<>/Font<>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 55 0 obj<> endobj 56 0 obj<> endobj 57 0 obj[/ICCBased 69 0 R] endobj 58 0 obj<> endobj 59 0 obj<> endobj 60 0 obj<> endobj 61 0 obj<>stream In a space-time region of space, there is a magnetic field in the equation E = * (3 imes 10*-2* T*) E = * (9 imes 10 *7 V m*-1*) * (*varepsilon_0 = 8.85 C2 N 1 M = 32.5* (;J m). (b) The magnetic field between the conductors can be found by applying Ampres law to the dashed path. Figure \(\PageIndex{3}\): Splitting of the energy levels for a I=1/2 (black dashed lines), I= 3/2 (blue dashed lines), and I=5/2 (red dashed lines . The energy stored in a magnetic field is equal to the work needed to produce a current through the inductor. The technology resulted from a decade of research and breakthrough engineering to produce and provide the cleanest energy power source for the demanding, power-hungry world. Again using the infinite solenoid approximation, we can assume that the magnetic field is essentially constant and given by B=0nIB=0nI everywhere inside the solenoid. This energy can be found by integrating the magnetic energy density, over the appropriate volume. and you must attribute OpenStax. \label{14.22}\]. The current revolution in the field of electromagnetic vibration energy harvester requires that both wireless sensor nodes and relevant power sources be cost- and size-optimized while ensuring that, during design/fabrication of the sensor's power sources, the power deliverable to the sensors be maximum. Magnetic field does not require any medium to propagate; it can propagate even in a vacuum. Like electric fields, magnetic fields can occupy completely empty space, and affect matter at a distance. 0000002663 00000 n then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a digital format, Also, the magnetic energy per unit length from part (a) is proportional to the square of the current. 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