The self-inductance depends on the geometric shape of the coil. An inductor is a coil of wire used in a circuit to provide inductance is an inductor.

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1 Self Inductance and Mutual Inductance Script Self-Inductance Consider a coil carrying a current i. The current in the coil produces a magnetic field B that varies from point to point in the coil. The magnetic field, B also varies with the current, i. The magnetic flux through the coil is proportional to Li. L is the proportionality constant called self-inductance. The self-inductance depends on the geometric shape of the coil. Inductors An inductor is a coil of wire used in a circuit to provide inductance is an inductor. An ideal inductor is one in which the wire forming the inductor has no resistance. The symbol for an inductor in a circuit is Inductance Inductance, L, of a solenoid is the ratio of flux to current. L = magnetic flux/ current This is self-inductance-the magnetic flux the solenoid creates in itself when there is a current. SI Unit of induction The SI unit of induction is the henry. As you can see from the last slide the henry must be equal to Webers/Amps. 1 henry = Wb/A = Tm 2 /A Solenoid

2 The self inductance of a solenoid, a long coil of wire, can be calculated directly. The magnetic flux through a solenoid of length, l, and N turns carrying a current of i Magnetic flux is equal to mu sub o N squared I times A divided by length. This is equal to mu sub o n squared I A l This means L inductance is equal to magnetic flux divided by current so inductance is equal to mu sub o n squared A l. Inductance Depends on Geometry As you see in the last slide the inductance of a solenoid depends only on its geometry. The constant µ o can be expressed in henrys per meter. µ o =4πx10-7 H/m Find the self inductance of a 100 turn solenoid of length 10 cm and area 5.0 cm 2. L = µ o n 2 Al L = 4πx10-6 (100 turns/m)(5x10-4 m 2 )(0.01m) L = 6.28 x 10-5 H EMF and Induction When the current in the circuit is changing, the magnetic flux due to the current is also changing, so an emf is induced in the circuit. Because the selfinductance of a circuit is constant, the change in flux is related to the change in current by the following equation: The derivative of flux with respect to time is equal to the derivative of inductance L times current I over dt. This equals L di/dt. EMF = negative derivative of flux/dt = -L di/dt

3 Inductor The self-induced emf is proportional to the inductance and the changing current. A coil or solenoid of many turns has a large self inductance and is called an inductor. The potential difference across an inductor is given by the following equation Delta V = emf Ir = -L di/dt - Ir Where r is the internal resistance of the inductor An inductor is made by wrapping 0.50 mm diameter of wire around a 4.4 mm diameter cylinder. What length cylinder has an inductance of 12 µh? View the following solution L = µ o N 2 A/l = µ o (l/d) 2 πr 2 /l = µ o πr 2 l/d 2 l= d 2 L/(µ o πr 2 ) = ( m 2 )(1.2x10-5 H)/[(4πx10-7 Tm/A)(0.0022m) 2 l = m = 15.7 cm Magnetic Energy An inductor stores magnetic energy much like a capacitor stores electrical energy. Mutual Inductance When two or more circuits are close together the magnetic flux through one circuit depends not only on the current in that circuit but also on the current in the nearby circuit or circuits. Mutual Inductance Mutual Inductance depends on the geometrical arrangement of the two circuits. If the circuits are far apart then the magnetic flux through circuit 2 will be small and the mutual inductance will be small. Magnetic flux of 2 on 1 equals Mutual inductance of 2 on 1 times i sub 1. Magnetic flux of 1 on 2 equals mutual inductance of 1 on 2 times i sub 2.

4 Concentric Solenoids Consider concentric two solenoids we can determine the mutual inductance for them. Let l be the length of both solenoids and let the inner solenoid have N 1 turns and r 1 radius. The outer solenoid has N 2 turns and r 2 radius. Let s first look at M 2,1 assuming the inner coil carries a current of I 1 and we will find the flux due to this current through the outer solenoid. Calculating Mutual Induction of Concentric Solenoids B1 = µ o (N 1 /l)i1 = µ o n 1 I 1 for r < r 1 Magnetic flux = N 2 B 1 (πr 1 2 )= n 2 lb 1 (πr 1 2 ) = µ o n 2 n 1 l(πr 1 2 )I 1 M 2 on 1 = flux 2 on 1 divided by I 1 = µ o n 2 n 1 lπr 1 2 EMF and Mutual Induction For any inductor having a self-inductance, L, the following equation applies. L = - emf sub l/(di/dt) We can write a mutual induction equation that is analogous to the selfinduction equation. M 1 on 2 = emf sub 2/ (di 1 /dt) EMF Induced in Close Coils EMF induced in either coil is proportional to the rate of change of current in the other coil. The emf of the second coil is related to the change in current of coil 1 and vice versa. Emf sub 2 = - M di 1 /dt Emf sub 1 = - M di 2 /dt Unit of Mutual Inductance The SI unit of mutual inductance is the henry, just like self-inductance.

5 Two solenoids one of radius 2.0 cm and 300 turns and the other has radius of 5.0 cm and 1000 turns. The solenoids are coaxial and both are each 25 cm long. Find their mutual inductance. See the solution below