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Inductance
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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
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Let’s start with: SELF-INDUCTANCE
The phenomenon of self-inductance was discovered by Joseph Henry in 1832 (Princeton University). Joseph Henry
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Self Inductance When current in the circuit changes, the flux changes also, and a self-induced voltage appears in the circuit
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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.
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(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
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From Faraday’s Law of Induction
(b) Definition that describes the behaviour of an inductor in a circuit From Faraday’s Law of Induction
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Two equivalent definitions of L
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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.
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Calculation of Self-Inductance
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The Self-Inductance of a Solenoid
n turns per unit length, radius R and the length of the solenoid is l
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Set up a current I, and we have a B field
Total number of turns is N=nl Flux through each turn
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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)
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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
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The Self-Inductance of a Toroid
b a b
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The Self-Inductance of a Toroid
dr h r b Consider an elementary strip of area hdr
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The Self-Inductance of a Toroid
Inductance – like capacitance – depends only on geometric factors
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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
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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:
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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
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The energy stored in the magnetic field of an inductor is analogous to that in the electric field of a capacitor
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Example: the energy stored in a solenoid
Energy per unit volume (magnetic energy density)
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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
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Review and Summary The self-inductance L is defined by the equations
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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
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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?
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(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.
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Mutual Inductance A changing current in loop 1 causes a changing flux in loop 2 inducing a voltage
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Mutual Inductance (1) The mutual inductance M21 is defined by the equation: (2) The mutual inductance M21 may also be defined by the equation:
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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)
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A Metal Detector Sinusoidally varying current
Parallel to the magnetic field of Ct
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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
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Revision – Ampere’s Circuital Law
Where is the line integral round a closed loop and Ienclosed is the current enclosed by the loop
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The B Field Due to a Long Straight Wire
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Next Installment Magnetic Materials
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