magnetic energy density formula

After the integration is carried out, we have a closed-form solution for part (a). "url": "", • xref The OpenStax name, OpenStax logo, OpenStax book The capacitance per unit length of the cable has already been calculated. © 1999-2020, Rice University. Textbook content produced by OpenStax is licensed under a Did you find apk for android? The circuit equations are thus, We intimated previously that the energy stored in an inductor is actually ε = permittivity. The general energy is: U = U E + U B. Energy Density = Energy × [Volume]-1. . • "item": The potential energy of a magnet of magnetic moment $${\displaystyle \mathbf {m} }$$ in a magnetic field $${\displaystyle \mathbf {B} }$$ 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: "position": 3, The inductance per unit length depends only on the inner and outer radii as seen in the result. 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, Maxwell’s 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. I$9���z�/ Q��b��J �3/�D^����9u��*��/�UP��!����lR���A���;4i�}�Y�7ß�W� �9�� "@id": "", Thus, the energy stored in a solenoid or the magnetic energy density times volume is equivalent to, \[U = u_m(V) = \dfrac{(\mu_0nI)^2}{2\mu_0}(Al) = \dfrac{1}{2}(\mu_0n^2Al)I^2. Its letter symbol is B. 0000005319 00000 n 0000004664 00000 n ��Q ��T;GP��z�����������u�.�� �Jn�����0�~6H J��������ʱ�%%��H�I�a�YeB(M2{.~X�m$��Vd偫�v�bd��?�8?�P��5�0��IJF���t8Ó�����"[�2&z��Q�b��u&�g��T��YG�Kwݮ_�@Or(�q�0J�&���ũ����8sn��[�䏢�JR@ed1%:��8M���� �,��-���q, F�lLʯ95�X��E�NE-AF�&���� m;� 蠢�m�MV�ѦY�+��LQsE�PaB�Z��\�X~�0Z[�p�����d�V!����Z����Xu�>%19kN�b���Pb]���wZ����wt��a��+�u�m� q��@�ɸ� � @��Ω�I"��/Z���8���or���P�����?�f���nӭ{ ���O�!��u�ٛl��n�� u0:�Zj�H� ��;� ]G�O/tx\�����T��(Ԕ�٠��� 0000015215 00000 n Let us now obtain an explicit formula for the energy stored in a magnetic field. We can see this by considering an arbitrary inductor through which a changing current is passing. “If one line of magnetic field passes normally through m2 area, the magnetic flux density, B, will be one Tesla, We know that 100 lines equal to 1 μWb. Determine the cross- sectional area of a toroid that has a flux of 0.5 Wb and a flux density of 25 T. We have the following formula to calculate cross-sectional area: $A=\frac{\varphi }{B}=\frac{0.5~Wb}{25~{}^{Wb}/{}_{{{m}^{2}}}}=2*~{{10}^{-2~}}{{m}^{2}}$. 51 26 Consider an ideal . • Inductance: L = m0n2A‘ • Magnetic field: B = m0nI • Potential energy: U = 1 2 LI2 = 1 2m0 B2(A‘) • Volume of solenoid interior: A‘ • Energy density of magnetic field: uB = U A‘ = 1 2m0 B2 tsl270 The self-inductance per unit length is determined based on this result and Equation \ref{14.22}. We have: U = energy density. . To understand where this formula comes from, let’s consider the long, cylindrical solenoid of the previous section. u = ε 0 2 E 2 + 1 2 μ 0 B 2. 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 (b) The magnetic field between the conductors can be found by applying Ampère’s law to the dashed path. The energy stored in the solenoid when a current flows through it There are 1010 turns of wire carrying a current of 9.56 A. © Sep 2, 2020 OpenStax. [ "article:topic", "authorname:openstax", "magnetic energy density", "license:ccby", "showtoc:no", "program:openstax" ], Creative Commons Attribution License (by 4.0), 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 Ampère’s law to the dashed circular path shown in Figure \(\PageIndex{1b}\). Energy Density Within Solenoid Energy is stored in the magnetic field inside the solenoid. Magnetic energy density = \(\frac {magnetic field squared} { 2 \times magnetic permeability }\) In the form of an equation, \(U_B\) = \(\frac {1}{2 \mu_0} B^2\) Magnetic energy and electrostatic potential energy are related by Maxwell's equations. 0000024211 00000 n "@type": "ListItem", \nonumber\] In the region outside the cable, a similar application of Ampère’s law shows that \(B = 0\), since no net current crosses the area bounded by a circular path where \(r > R_2\). not be reproduced without the prior and express written consent of Rice University. While the total amount of the flux produced by a magnet is important, we are more interested in how dense or concentrated, the flux is per unit of cross-sectional area. After the integration is carried out, we have a closed-form solution for part (a). 0000000816 00000 n covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may To understand where this formula comes from, let’s consider the long, cylindrical solenoid of the previous section. endstream endobj 62 0 obj<> endobj 63 0 obj<> endobj 64 0 obj<> endobj 65 0 obj<> endobj 66 0 obj<>stream each coil is connected to its own battery. } } ] In cgs, the energy density contained in a magnetic field B is (1) and in MKS is given by (2) "@id": "", "@context": "", "@type": "BreadcrumbList", Let us now obtain an ��w�G� xR^���[�oƜch�g�`>b���$���*~� �:����E���b��~���,m,�-��ݖ,�Y��¬�*�6X�[ݱF�=�3�뭷Y��~dó ���t���i�z�f�6�~`{�v���.�Ng����#{�}�}��������j������c1X6���fm���;'_9 �r�:�8�q�:��˜�O:ϸ8������u��Jq���nv=���M����m����R 4 � Home » Magnetism » Magnetic Flux Density | Definition and Formula { I am very happy that I came across this during my hunt for something concerning this. The magnetic energy is calculated by an integral of the magnetic energy density times the differential volume over the cylindrical shell. 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}. Calculate the flux density if the core diameter is 4 cm. 0000001430 00000 n startxref In fact, the above formula is valid for any two inductively coupled circuits. 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. To increase the inductance, we could either increase the outer radius (\(R_2\)) or decrease the inner radius (\(R_1\)). 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