Today in Physics 1: inductance Mutual inductance elf inductance nductors in circuits: and circuits. it Energy stored in inductors 15 November 01 Physics 1, Fall 01 1 Mutual inductance onsider again a transformer which couples magnetic flux perfectly between two coils. uppose the transformer coils are long solenoids, with crosssection area A and length. Apply a sinusoidallyvarying (i.e. A) voltage to the primary side. For the secondary coil, NP BPA AP B NPNA per turn of NB P the coil (from Giancoli) 15 November 01 Physics 1, Fall 01 f instead we impose a current on the secondary side, the flux on the primary side is N BA A BP NP N A NPBP Evidently the two coils have an symmetrical relation between the current in one and the flux in the other. Define: NPBP NB NPNA M P (from Giancoli) 15 November 01 Physics 1, Fall 01 3 (c) University of ochester 1
The mutual inductance M for the transformer depends only upon the geometrical factors of the coils and the permeability of the magnetic material which conducts the flux. This result turns out generally to be obtained for circuits which can induce currents in each other, since the B generated by a current depends linearly on in most cases. (from Giancoli) 15 November 01 Physics 1, Fall 01 4 n general, for two loops with number of turns N1 and N and currents 1 and, NB M11 N1B1 M1 where the mutual inductances depend only on loop geometry and permeability of the medium between them. Furthermore, it can be proven that, as we saw for the transformer, M1 M1 M 1 N 1 N 15 November 01 Physics 1, Fall 01 5 The main implication of mutual inductance is that the EMF induced in one circuit due to a time dependent fllux from another circuit is proportional to the time derivative of the current in that other circuit: db d 1 N M db1 d 1 N1 M since M depends only on geometry and permeability, not upon time. 1 N 1 N 15 November 01 Physics 1, Fall 01 6 (c) University of ochester
elf inductance t s easiest to imagine this sort of inductive circuit coupling between two circuits, but of course it happens with just one: External voltage applied to circuit drives current, current generates B which has flux through the circuit, induced EMF appears in the circuit. For a long solenoid (N turns, length, loop area A, current ), N B BA A N NB N A elf A inductance db d N 15 November 01 Physics 1, Fall 01 7 elf inductance (continued) Other sorts of loops will also have self inductance, though it isn t usually as easy to calculate as it is for the solenoid. Main implication: we have another lumped circuit element, along with voltage sources, resistors, and capacitors: the inductor. 1 Q n honor h of f the noble V olenoid, inductors are represented as coils in circuit diagrams, whether V solenoidal or not. n honor of enz, self d d Q inductance is given V the symbol. 15 November 01 Physics 1, Fall 01 8 nductors (selfinductances) in circuits: implest example: an circuit. witch is closed at t = 0. What is the current as a function of time? As usual we use Kirchhoff s rules. No nodes and only one loop: d V 0, or d V, or t t d V 0 0 V 15 November 01 Physics 1, Fall 01 9 (c) University of ochester 3
(continued) expect you re tired of doing this integral so often: V q dq d V V q t t V dq t q V t V ln V V 15 November 01 Physics 1, Fall 01 10 Exponentiate both sides, as (continued) usual: t V e t V V t 1 e t ike the circuit, this one has a time constant: t t max 1 e max V (t) max 1 1 e 0 4 6 8 10 t (/) 15 November 01 Physics 1, Fall 01 11 (continued) Note the strong resemblance of current in the circuit to charge in the circuit. (t) max 1 1 e 0 4 6 8 10 t (/) Q(t) Q max 1 1 e (t) 1 e max 0 4 6 8 10 t () 0 4 6 8 10 t () 15 November 01 Physics 1, Fall 01 1 (c) University of ochester 4
Next simplest: add a capacitor. witch is closed at t = 0. What is the current as a function of time? Again no nodes and only one loop: Q d 0 dq 1 Q 0 This should look familiar from PHY 11: it s the equation for undamped simple harmonic motion, for which the solutions are sinusoidal. 15 November 01 Physics 1, Fall 01 13 (continued) uppose the solutions to be of the form Q Q 0 cos t, where 0, and are constants. Then Q QQ0 cos t dq Q sin t 0 dq Q 0 cost Q 0 cos t 1 Q0 cost0 15 November 01 Physics 1, Fall 01 14 (continued) or 1 1 0 The current spontaneously oscillates with an angular frequency of 1. 15 November 01 Physics 1, Fall 01 15 (c) University of ochester 5
Energy in d We have shown above that V, so d P U W P 1 d 0 1 Q ompare to U for capacitors: in circuits, energy oscillates back and forth from inductor to capacitor. 15 November 01 Physics 1, Fall 01 16 (c) University of ochester 6