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Lecture 22. Inductance. Magnetic Field Energy. Outline: Self-induction and self-inductance. Inductance of a solenoid. The energy of a magnetic field. Alternative definition of inductance. Mutual Inductance. 1 - CH2 exam average ~54 %. - To the right are VERY tentative correlations between percentage scores and letter grades in Physics 227. The instructors reserve the right to adjust these boundaries (both up and down) before final grades are assigned at the end of the semester. A ≥ 85 80 ≤ B+ < 85 70 ≤ B < 80 65 ≤ C+ < 70 53 ≤ C < 65 40 ≤ D < 53 40 < F 2 Maxwell’s Equations Gauss’s Law: charges are sources of electric field (and a non-zero net electric field flux through a closed surface); field lines begin and end on charges. 𝑄𝑒𝑒𝑒𝑒 𝜀0 � 𝐸 ∙ 𝑑𝐴⃗ = � 𝐵 ∙ 𝑑𝐴⃗ = 0 � 𝐸 ∙ 𝑑𝑙⃗ = − � 𝐵 ∙ 𝑑𝑙⃗ = 𝜇0 � 𝑠𝑠𝑠𝑠 No magnetic monopoles; magnetic field lines form closed loops. 𝑠𝑠𝑠𝑠 Faraday’s Law of electromagnetic induction; a time-dependent ΦB generates E. 𝑙𝑜𝑜𝑜 𝑑 � 𝐵 ∙ 𝑑𝐴⃗ 𝑑𝑑 𝑠𝑢𝑢𝑢 𝑑Φ𝐵 ℰ=− 𝑑𝑑 Generalized Ampere’s Law; B is produced by both currents and time-dependent ΦE. 𝑙𝑜𝑜𝑜 𝑠𝑢𝑢𝑢 𝐽⃗ + 𝜖0 The force on a (moving) charge. 𝐹⃗ = 𝑞 𝐸 + 𝑣⃗ × 𝐵 𝜕𝐸 ∙ 𝑑𝐴⃗ 𝜕𝜕 3 Induced E.M.F. and its consequences 𝑑Φ𝐵 𝐹𝐹𝐹𝐹𝐹𝐹𝐹: ℰ = − 𝑑𝑑 1. 2. 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝑩 𝒕 3. 𝐼 𝑡 𝐼 𝑡 𝑩 𝒕 8 (Self) Inductance Variable battery 𝐼 𝑡 𝑩 𝒕 The back e.m.f. : 𝓔 =− Inductance: (or self-inductance) 𝑑Φ𝐵 𝐹𝐹𝐹𝐹𝐹𝐹𝐹: ℰ = − 𝑑𝑑 The flux depends on the current, so a wire loop with a changing current induces an “additional” e.m.f. in itself that opposes the changes in the current and, thus, in the magnetic flux (sometimes called the back e.m.f.). 𝒅𝜱𝑩 𝒅𝒅 = −𝑳 𝒅𝒅 𝒅𝒅 𝜱 𝑳≡ 𝑰 𝐿 – the coefficient of proportionality between Φ and 𝐼, purely geometrical quantity. Units: T⋅m2/A = Henry (H) The self-inductance L is always positive. This definition applies when there is a single current path (see below for an inductance of a system of distributed currents 𝐽 𝑟 ). Inductance as a circuit element: 9 Inductance of a Long Solenoid Long solenoid of radius r and length l with the total number of turns 𝑁: 𝐿≡ Φ𝐵 𝐼 l Note that The magnetic flux generated by current 𝐼: 𝑁 𝜇0 𝑁 2 𝐼 2 Φ𝐵 = 𝐵𝜋𝑟 𝑁 = 𝐵 = 𝜇0 𝐼 = 𝜋𝑟 = 𝜇0 𝑛2 𝐼 l 𝜋𝑟 2 l l 2 Φ𝐵 𝜇0 𝑁 2 2 𝐿= = 𝜋𝑟 = 𝜇0 𝑛2 l 𝜋𝑟 2 𝐼 l (n – the number of turns per unit length) - the inductance scales with the solenoid volume l 𝜋𝑟 2 ; 𝑁 - the inductance scales as N2: B is proportional to l and the net flux ∝ N. Let’s plug some numbers: l =0.1m, N=100, r=0.01m 100 𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 = 4𝜋 ∙ 10−7 ∙ 0.1 How significant is the induced e.m.f.? For 𝑑𝑑 ~104 𝐴/𝑠 (let’s say, ∆𝐼~ 1 mA in 0.1 µs) 𝑑𝑑 2 ∙ 0.1 ∙ 𝜋 0.01 2 ≈ 40𝜇𝜇 𝑑𝑑 4 ∙ 10−5 𝐻 × 1 ∙ 104 𝐴 ℰ =𝐿 = = 0.4𝑉 𝑑𝑑 𝑠 10 Inductance vs. Resistance Inductor affects the current flow only if there is a change in current (di/dt ≠ 0). The sign of di/dt determines the sign of the induced e.m.f. The higher the frequency of current variations, the larger the induced e.m.f. 13 Energy of Magnetic Field To increase a current in a solenoid, we have to do some work: we work against the back e.m.f. Let’s ramp up the current from 0 to the final value I (𝑖 𝑡 is the instantaneous current value): V 𝑖 𝑡 𝑖 𝑡 𝑉 𝑡 = 𝐿 𝑑𝑑 = 𝑃 𝑡 𝑑𝑑 The net work to ramp up the current from 0 to 𝐼 : 𝑑𝑑 𝑡 𝑑𝑑 𝑃 𝑡 =𝑉 𝑡 𝑖 𝑡 =𝐿 𝑑𝑑 𝑡 𝑖 𝑡 𝑑𝑑 P(power) - the rate at which energy is being delivered to the inductor from external sources. 𝑓 𝐼 1 2 𝑈 = � 𝑃 𝑡 𝑑𝑑 = � 𝐿𝐿 ∙ 𝑑𝑑 = 𝐿𝐼 2 𝑖 0 𝟏 𝟐 𝑼 = 𝑳𝑰 𝟐 This energy is stored in the magnetic field created by the current. 𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 = 𝜇0 𝑛2 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 1 𝐵2 2 2 𝑈𝐵 = 𝜇0 𝑛 𝑣𝑣𝑣𝑣𝑣𝑣 ∙ 𝐼 = 𝐵 = 𝜇0 𝑛𝑛 = ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐵 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 2𝜇0 2 The energy density in a magnetic field: 𝑩𝟐 𝒖𝑩 = 𝟐𝝁𝟎 (compare with 𝑢𝐸 = This is a general result (not solenoid-specific). 𝜀0 𝐸 2 2 ) 14 Quench of a Superconducting Solenoid MRI scanner: the magnetic field up to 3T within a volume ~ 1 m3. The magnetic field energy: 𝑈 = 𝑣𝑣𝑣𝑣𝑣𝑣 ∙ 𝑢𝐵 = 1𝑚 The capacity of the LHe dewar: ~2,000 liters. 3 3𝑇 2 𝑊𝑊 2 ∙ 4𝜋 ∙ 10−7 𝐴∙𝑚 ≈ 4𝑀𝑀 The heat of vaporization of liquid helium: 3 kJ/liter. Thus, ~ 1000 liters will be evaporated during the quench. Magnet quench: http://www.youtube.com/watch?v=tKj39eWFs10&feature=related 15 Another (More General) Definition of Inductance Alternative definition of inductance: 1 2 𝐿𝐼 = 2 � 𝑎𝑙𝑙 𝑠𝑠𝑠𝑠𝑠 𝐵2 𝑑𝜏 2𝜇0 𝑳≡ 𝟐𝑼𝑩 𝑰𝟐 Advantage: it works for a distributed current flow (when it’s unclear which loop we need to consider for the flux calculation). 16 Example: inductance per unit length for a coaxial cable 𝐼 𝑏 𝑎 𝐼 Uniform current density in the central conductor of radius a: 𝑏 1 𝑈𝐵 = � 𝐵 𝑟 2𝜇0 0 2 𝑑𝜏 𝐵 𝑟 = Straight Wires: Do They Have Inductance? Using 𝑳 = 𝟐𝑼𝑩 /𝑰𝟐 : 𝑏 𝜇0 𝐼 1 𝑈𝐵 = � 2𝜋𝜋 2𝜇0 𝑎 2 𝑏 𝜇0 𝐼 1 𝑑𝜏 = � 2𝜋𝜋 2𝜇0 𝑎 Using 𝑳 = 𝜱/𝑰: 𝜇 𝐼 ~l 𝐵 ≈ 0 2𝜋𝜋 l l Φ ≈ l� 𝑎 2 𝜇0 𝐼 ,𝑎 < 𝑟 < 𝑏 2𝜋𝜋 𝜇0 𝐼𝑟 , 2𝜋𝑎2 𝑟<𝑎 𝜇0 𝐼 2 l ∙ l ∙ 2𝜋𝜋𝜋𝜋 ≈ (𝑏~l ) ≈ ∙ l ∙ 𝑙𝑙 4𝜋 𝑎 𝜇0 𝐼 𝝁𝟎 𝑰 l 𝑑𝑑 = ∙ l ∙ 𝒍𝒍 2𝜋𝜋 𝟐𝝅 𝒂 at home 𝐿= 2𝑈𝐵 𝐼2 l 𝝁𝟎 𝑳𝑯 ≈ l ∙ 𝒍𝒍 𝟐𝝅 𝒂 𝜇0 l l −7 𝐿𝐻 ≈ l ∙ 𝑙𝑙 ≈ 2 ∙ 10 l 𝑚 ∙ 𝑙𝑙 2𝜋 𝑎 𝑎 l = 1𝑐𝑐 𝐿 ≈ 10−9 𝐻 17 Inductance vs. Capacitance 𝐿= Φ𝐵 2𝑈𝐵 = 2 𝐼 𝐼 1 2 1 Φ𝐵 2 𝑈𝐵 = 𝐿𝐼 = 2 2 𝐿 𝐵2 𝑈𝐵 = ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐵 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 2𝜇0 𝐶= 𝑄 2𝑈𝐸 = 2 𝑉 𝑉 1 2 1 𝑄2 𝑈𝐸 = 𝐶𝑉 = 2 2 𝐶 𝜖0 𝐸 2 𝑈𝐸 = ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑢𝐸 ∙ 𝑣𝑣𝑣𝑣𝑣𝑣 2 18 Mutual Inductance Let’s consider two wire loops at rest. A time-dependent current in loop 1 produces a time-dependent magnetic field B1. The magnetic flux is linked to loop 1 as well as loop 2. Faraday’s law: the time dependent flux of B1 induces e.m.f. in both loops. loop 2 𝑩𝟏 The e.m.f. in loop 2 due to the time-dependent I1 in loop 1: ℰ2 = − 𝐼1 loop 1 Primitive “transformer” 𝑑Φ1→2 𝑑𝑑 Φ1→2 = 𝑀1→2 𝐼1 The flux of B1 in loop 2 is proportional to the current I1 in loop 1. The coefficient of proportionality 𝑀1→2 (the so-called mutual inductance) is, similar to L, a purely geometrical quantity; its calculation requires, in general, complicated integration. Also, it’s possible to show that 𝑀1→2 = 𝑀2→1 = 𝑀 so there is just one mutual inductance 𝑀. 𝑀 can be either positive or negative, depending on the choices made for the senses of transversal about loops 1 and 2. Units of the (mutual) inductance – Henry (H). ℰ2 = −𝑀 𝜕𝐼1 𝜕𝜕 𝑀= Φ1→2 𝐼1 19 Calculation of Mutual Inductance l 𝑀1→2 = 𝑀2→1 = 𝑀 Calculate M for a pair of coaxial solenoids (𝑟1 = 𝑟2 = 𝑟, l 1 = l 2 = l, 𝑛1 and 𝑛2 - the densities of turns). 𝐵1 = 𝜇0 𝑛1 𝐼1 the flux per one turn 2 𝑁2 Φ 𝜇 𝑁 𝑁 𝜋𝑟 0 1 2 𝑀= = 𝑁2 𝜇0 𝑛1 𝜋𝑟 2 = l 𝐼1 2 𝜋𝑟 2 𝜇 𝑁 0 𝐿 = 𝜇0 𝑛2 l 𝜋𝑟 2 = l Experiment with two coaxial coils. Φ = 𝐵1 𝜋𝑟 2 = 𝜇0 𝑛1 𝐼1 𝜋𝑟 2 𝑀= 𝐿1 𝐿2 The induced e.m.f. in the secondary coil is much greater if we use a ferromagnetic core: for a given current in the first coil the magnetic flux (and, thus the mutual inductance) will be ~𝐾𝑚 times greater. 𝑑𝐼1 𝑑𝐼1 ℰ2 = 𝑀∗ = 𝐾𝑚 𝑀 𝑑𝑑 𝑑𝑑 no-core mutual inductance 20 Next time: Lecture 23. RL and LC circuits. §§ 30.4 - 6 21 Appendix I: Inductance of a Circular Wire Loop 2𝑎 Example: estimate the inductance of a circular wire loop with the radius of the loop, b, being much greater than the wire radius a. 𝑏 The field dies off rapidly (~1/r) with distance from the wire, so it won’t be a big mistake to approximate the loop as a straight wire of length 2πb, and integrate the flux through a strip of the width b: b 𝜇0 𝐼 𝐵≈ 2𝜋𝜋 𝑏 Φ ≈ 2𝜋𝜋 � 2πb 𝑎 𝐿 𝑏 = 1𝑚, 𝑎 = 𝜇0 𝐼 𝑏 𝑑𝑑 = 𝜇0 𝑏𝐼 ∙ 𝑙𝑙 2𝜋𝜋 𝑎 10−3 𝑚 ≈ 4𝜋 ∙ Note that we cannot assume that the wire radius is zero: 𝑏 𝜇0 𝐼 𝑏 Φ~ � 𝑑𝑑 ~𝑙𝑙 2𝜋𝜋 "0" - diverges! 10−7 𝐿= Φ 𝑏 ≈ 𝜇0 𝑏 ∙ 𝑙𝑙 𝐼 𝑎 𝑊𝑊 1𝑚 ∙ 𝑙𝑙103 ≈ 10𝜇𝜇 𝐴∙𝑚 What to do: introduce some reasonably small non-zero radius. "0" 22