Inductance

Inductance Learning Objectives Definition and Calculation of Self-Inductance To obtain an expression for the Energy Stored by an Inductor Definition and Calculation of Mutual-Inductance

Let’s start with: SELF-INDUCTANCE The phenomenon of self-inductance was discovered by Joseph Henry in 1832 (Princeton University). Joseph Henry 1797-1878

Self Inductance When current in the circuit changes, the flux changes also, and a self-induced voltage appears in the circuit

L is the self-inductance of the coil. I constant, e= 0 I increasing or decreasing , e = Vab>0 b a L is the self-inductance of the coil.

(a) Definition used to find L Suppose a current I in a coil of N turns causes a flux B to thread each turn The self-inductance L is defined by the equation

From Faraday’s Law of Induction (b) Definition that describes the behaviour of an inductor in a circuit From Faraday’s Law of Induction

Two equivalent definitions of L

This is called the henry (H) SI unit for inductance V s A-1 This is called the henry (H) If a current changing by 1A/s is to generate 1V, the inductance is 1H.

Calculation of Self-Inductance

The Self-Inductance of a Solenoid n turns per unit length, radius R and the length of the solenoid is l

Set up a current I, and we have a B field Total number of turns is N=nl Flux through each turn

number of turns per unit length) The inductance does not depend on current or voltage, it is a property of the coil. (length, width, and number of turns per unit length)

Find the self-inductance of a solenoid of length 10 cm, area 5 cm2, and 100 turns. n = 100/0.1 = 1000 turns/m At what rate must the current in the solenoid change to induce a voltage of 20 V? Answer: 3.18  105 A/s

The Self-Inductance of a Toroid b a b

The Self-Inductance of a Toroid dr h r b Consider an elementary strip of area hdr

The Self-Inductance of a Toroid Inductance – like capacitance – depends only on geometric factors

Unit of 0 is H m-1 0 = 4  10-7 H m-1 From the worked examples it can be seen that: Unit of 0 is H m-1 0 = 4  10-7 H m-1 0 = 4  10-7 wb/Am

The Energy Stored by an Inductor I increasing a b (Faraday’s law in disguise)  The energy dU supplied to the inductor during an infinitesimal time interval dt is:

The Energy Stored by an Inductor The total energy U supplied while the current increases from zero to a final value I is This energy is stored in the magnetic field

The energy stored in the magnetic field of an inductor is analogous to that in the electric field of a capacitor

Example: the energy stored in a solenoid Energy per unit volume (magnetic energy density)

The equation is true for all magnetic field configurations MAGNETIC ENERGY DENSITY IN A VACUUM The equation is true for all magnetic field configurations Compare with the energy density in an electric field

Review and Summary The self-inductance L is defined by the equations

Review and Summary An inductor with inductance L carrying current I has potential energy This potential energy is associated with the magnetic field of the inductor. In a vacuum, the magnetic energy per unit volume is

How would the self-inductance of a solenoid be changed if the same length of wire were wound onto a cylinder of the same diameter but twice the length? twice as much wire were wound onto the same cylinder? the same length of wire were wound onto a cylinder of the same length but twice the diameter?

(a) Since the diameter does not change, the number of turns and the area A remain constant. However, n2 is diminished by a factor of 4 and l is increased by a factor of 2. Thus L is reduced by a factor of 2. (b) Using twice as much wire and making no other change, n2 and L are increased by a factor of 4. (c) With twice the diameter, n2 is reduced by a factor of 4, but A is increased by the same factor; L is unchanged.

Mutual Inductance A changing current in loop 1 causes a changing flux in loop 2 inducing a voltage

Mutual Inductance (1) The mutual inductance M21 is defined by the equation: (2) The mutual inductance M21 may also be defined by the equation:

Mutual Inductance It can be proved that the same value is obtained for M if one considers the flux threading the first loop when a current flows through the second loop (mutual inductance) (mutually induced voltages)

A Metal Detector Sinusoidally varying current Parallel to the magnetic field of Ct

Review and Summary If two coils are near each other, a changing current in either coil can induce a voltage in the other. This mutual induction phenomenon is described by where M (measured in henries) is the mutual inductance for the coil arrangement

Revision – Ampere’s Circuital Law Where is the line integral round a closed loop and Ienclosed is the current enclosed by the loop

The B Field Due to a Long Straight Wire

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