Energy Stored in Magnetic Fields: Inductors and Magnetic Fields, Maxwell's Term, Study notes of Electromagnetism and Electromagnetic Fields Theory

Download Energy Stored in Magnetic Fields: Inductors and Magnetic Fields, Maxwell's Term and more Study notes Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! 1 PHY481 - Lecture 24: Energy in the magnetic field, Maxwell’s term Griffiths: Chapter 7 Energy stored in inductors An external voltage source is used to provide the energy required to establish a magnetic field in an inductor. The rate at which work is done by the external source is, P = dW dt = dU dt = V I (1) We have shown that the voltage across an inductor is −LdI/dt, so we have, dU dt = LI dI dt (2) The total energy stored in an inductor is the integral of the work so that, U = ∫ ∞ 0 dU dt dt = ∫ i 0 LI dI dt dt = 1 2 Li2 (3) where i is the final current. Energy stored in magnetic fields This energy is stored in the magnetic field of the inductor. By considering a solenoid we can find the energy density in the magnetic field, through U = 1 2 Li2 = 1 2 µ0n 2Ali2 = 1 2µ0 B2Al (4) using B = µ0ni, L = µ0N2A/l and Al = volume, the energy density in the magnetic field is then found to be, u = 1 2µ0 B2. (5) Energy in terms of current sources Consider the flux in a single loop, φB = ∫ ~B · d~a = ∫ (~∇∧ ~A).d~a = ∮ ~A · d~l = Li (6) To find an expression for the energy in terms of distributed current sources, we write, U = 1 2 i ∫ ~A · d~l→ 1 2 ∫ ~A ·~idl→ 1 2 ∫ ~A ·~jdτ. (7) Where the last expression on the right hand side may be considered to be a postulate. To show that it is the same as the expression in terms of the energy density in the magnetic field, we write, 1 2 ∫ ~A ·~jdτ = 1 2µ0 ∫ ~A · (~∇∧ ~B)dτ (8) where we used Ampere’s law, ~∇∧ ~B = µ0~j. Now use the vector identity, ~∇ · ( ~A ∧ ~B) = ~A · (~∇∧ ~B)− ~B · (~∇∧ ~A) (9) so that, 1 2µ0 ∫ ~A · (~∇∧ ~B)dτ = − 1 2µ0 ∫ ~∇ · ( ~A ∧ ~B)dτ + 1 2µ0 ∫ ~B · (~∇∧ ~A)dτ = 1 2µ0 ∫ B2dτ (10) where the term ∫ ~∇ · ( ~A ∧ ~B)dτ is removed by using the divergence theorem to convert it to a surface integral and then taking the volume to infinity. 2 The energy to set up a magnetic field may then be expressed in two ways, 1 2 ∫ ~A ·~jdτ = 1 2µ0 ∫ all space B2dτ (11) Recall that the energy to set up an electric charge distribution is 1 2 ∫ ρV dτ = 0 2 ∫ all space E2dτ (12) which look very similar. The energy stored in magnetic fields can be large and has been proposed as an alternative to batteries, provided superconducting wires are used in order to minimize the resistive losses. Large amounts of energy are stored in the earth’s magnetic field and in the magnetic fields of galaxies. Example: Comparison of the energy stored in the earth’s electric and magnetic fields The energy stored in the electric field of the earth is, Uelectric ≈ V olume ∗ 0 2 (110V/m)2 = (4π(6400km)22km) 0 2 (110V/m)2 ≈ 4 ∗ 1010J (13) Is this a lot of energy? One gallon of gasolene has energy content 1.26 × 108J , so its relatively small In the above we use the fact that the earth’s electric field reduces to about one half of its sea level value at an altitude of 2km. There are considerable variations from place to place on the earth’s surface and this is a rough average value, but good enough for an estimate correct to an order of magnitude or so. The magnetic field of the earth extends to far greater distances than the earth’s electric field, so we have to take into account the decay of the field with distance. The magnetic field at the earth surface varies in magnitude, from place to place, in the range 20µT to 70µT . Lets take 50µT as a reasonable estimate. We may then take the magnitude of the earth field with distance to be approximately 50µT (Re/r)3, where Re = 6400km is the radius of the earth. An estimate of the energy stored in the earth’s magnetic field is then, Umagnetic ≈ 1 2µ0 ∫ ∞ Re (50µT )2R6e 4πr2 r6 dr = 2π(50µT )2R3e 3µ0 ≈ 1017J, (14) which is much larger than the energy stored in the electric field. A typical power station is a giga watt, and the number of seconds in a year is about 3× 107 so a power station running for about 3 years could generate the energy stored in the earth’s field. Maxwell term Maxwell noticed that there is a logical inconsistency in Ampere’s law that is easily illustrated in the case of charging capacitor. In that case Ampere’s law may be used to calculate the magnetic field near the wire and using a simple closure of the contour we get the usual result B(r) = µoi2πs . However we may use any area to calculate the enclosed current, for example a contour that passes between the plates of the capacitor. In that no current is enclosed so Ampere’s law says that the magnetic field is zero. Therefore Ampere’s law cannot apply to the case of a time varying current. Maxwell resolved this difficulty by adding a new term which includes the effect of the electric field which builds up between the capacitor plates. His idea was to relate this electric field to the current flowing the circuit. From Gauss’ law, we have, dφE dt = 1 0 dq dt (15) or the ”Maxwell displacement current” is, id = dq dt = 0 dφE dt (16) This relates the current flowing into the capacitor to the electric field between the plates so that Ampere’s law is modified to, ∮ ~B · d~l = µ0(i+ id) = µ0i+ µ00 dφE dt (17)