Self-inductance Self-inductance occurs when the changing flux through a circuit arises from the circuit itself Self-inductance occurs when the changing flux through a circuit arises from the circuit itself As the current increases, the magnetic flux through a loop due to this current also increasesAs the current increases, the magnetic flux through a loop due to this current also increases The increasing flux induces an emf that opposes the currentThe increasing flux induces an emf that opposes the current As the magnitude of the current increases, the rate of increase lessens and the induced emf decreasesAs the magnitude of the current increases, the rate of increase lessens and the induced emf decreases This opposing emf results in a gradual increase of the currentThis opposing emf results in a gradual increase of the current

Self-inductance cont The self-induced emf must be proportional to the time rate of change of the current The self-induced emf must be proportional to the time rate of change of the current L is a proportionality constant called the inductance of the deviceL is a proportionality constant called the inductance of the device The negative sign indicates that a changing current induces an emf in opposition to that changeThe negative sign indicates that a changing current induces an emf in opposition to that change

Self-inductance, final The inductance of a coil depends on geometric factors The inductance of a coil depends on geometric factors The SI unit of self-inductance is the Henry The SI unit of self-inductance is the Henry 1 H = 1 (V · s) / A 1 H = 1 (V · s) / A

For infinitely long solenoid B =  o nI n: number of turns/m

Increase current through the coil from 0 to  i in  t. Winding density: n Loop area: A Length of coil: ℓ Causes flux change for each loop from  = 0 to  = AB where B =  o n  i V ind = - n ℓ  /  t = - n ℓ A(  o n)  i/  t = - n 2  o A ℓ (  i / t) L: self inductance Purely geometrical quantity [L] = Henry

r = 1 cm ℓ = 2.5 cm n = 10 turns/cm = 0.01 m = m = 1000 turns/m L = n 2  o A ℓ = (1000) 2 (4 x )(x )0.025 = 9.9 x H = 9.9 H

Example 24.5 A steady current of 5.0 A is flowing through a coil with 2 H Self inductance. The current is suddenly stopped in 0.01 s. How large an emf induced in the coil? V ind = -L (i/t) i = i f – i i = -5 A = - (2.0) (-5/0.01) = 1000 V

Inductors in an AC Circuit Consider an AC circuit with a source and an inductor Consider an AC circuit with a source and an inductor The current in the circuit is impeded by the back emf of the inductor The current in the circuit is impeded by the back emf of the inductor The voltage across the inductor always leads the current by 90° (T/4) The voltage across the inductor always leads the current by 90° (T/4)

AC A V v= v o sin(  t) v i v = -L  i/  t i = -i o cos(  t) v o sin(  t) =  Li o sin(  t) v o =  Li o i lags 90 deg behind v.

Inductive Reactance and Ohm’s Law The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by X L = L=2ƒLX L = L=2ƒL When ƒ is in Hz and L is in H, X L will be in ohms When ƒ is in Hz and L is in H, X L will be in ohms Ohm’s Law for the inductor Ohm’s Law for the inductor V = I X LV = I X L

Inductor in a Circuit Inductance can be interpreted as a measure of opposition to the rate of change in the current Inductance can be interpreted as a measure of opposition to the rate of change in the current Remember resistance R is a measure of opposition to the currentRemember resistance R is a measure of opposition to the current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current Therefore, the current doesn’t change from 0 to its maximum instantaneouslyTherefore, the current doesn’t change from 0 to its maximum instantaneously

RL Circuit When the current reaches its maximum, the rate of change and the back emf are zero When the current reaches its maximum, the rate of change and the back emf are zero The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value

RL Circuit, cont The time constant depends on R and L The time constant depends on R and L The current at any time can be found by The current at any time can be found by

Transformer Iron Core i1i1 V2V2 N1N1 N2N2 = - (  i 1 /  t)M 21 Mutual inductance AC To ch. circuit

The RLC Series Circuit The resistor, inductor, and capacitor can be combined in a circuit The resistor, inductor, and capacitor can be combined in a circuit The current in the circuit is the same at any time and varies sinusoidally with time The current in the circuit is the same at any time and varies sinusoidally with time

Current and Voltage Relationships in an RLC Circuit The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the inductor leads the current by 90° The instantaneous voltage across the inductor leads the current by 90° The instantaneous voltage across the capacitor lags the current by 90° The instantaneous voltage across the capacitor lags the current by 90°

AC A i= i o sin(  t) VLVL VCVC v = v c + v L + v R VRVR i vR vL vC i= i o sin(  t) v o = Xi o sin(  t)Ri o i lags 90 deg behind v. cos(  t)  Li o - cos(  t)(1/  C)i o V lags 90 deg behind i.

v = v c + v L + v R = -(1/  C)i o cos(  t) +  Li o cos(  t) + Ri o sin(  t) = {(  L-1/  C)i o } cos(  t) + Ri o sin(  t) = {(  L-1/  C) 2 + R 2 } 1/2 (i o ) sin(  t+  ) X RLC v o = Xi o X RLC = {(  L-1/  C) 2 + R 2 } 0.5   = 1/(LC) 0.5 R i o = v o /X RLC Resonance: L = 1/C RLC Series Resonance Circuit

AC A i= i o sin(  t) V vovo RLC Parallel Resonance Circuit  = 1/(LC) 0.5

Power in an AC Circuit No power losses are associated with capacitors and pure inductors in an AC circuit No power losses are associated with capacitors and pure inductors in an AC circuit In a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuitIn a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuit In an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuitIn an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit

Power in an AC Circuit, cont The average power delivered by the generator is converted to internal energy in the resistor The average power delivered by the generator is converted to internal energy in the resistor P av = IV R = IV cos P av = IV R = IV cos  cos  is called the power factor of the circuitcos  is called the power factor of the circuit Phase shifts can be used to maximize power outputs Phase shifts can be used to maximize power outputs

Resonance in an AC Circuit Resonance occurs at the frequency, ƒ o, where the current has its maximum value Resonance occurs at the frequency, ƒ o, where the current has its maximum value To achieve maximum current, the impedance must have a minimum valueTo achieve maximum current, the impedance must have a minimum value This occurs when X L = X CThis occurs when X L = X C

Resonance, cont Theoretically, if R = 0 the current would be infinite at resonance Theoretically, if R = 0 the current would be infinite at resonance Real circuits always have some resistanceReal circuits always have some resistance Tuning a radio Tuning a radio A varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be receivedA varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received Metal Detector Metal Detector The portal is an inductor, and the frequency is set to a condition with no metal presentThe portal is an inductor, and the frequency is set to a condition with no metal present When metal is present, it changes the effective inductance, which changes the current which is detected and an alarm soundsWhen metal is present, it changes the effective inductance, which changes the current which is detected and an alarm sounds

Simple Radio

Summary of Circuit Elements, Impedance and Phase Angles

James Clerk Maxwell Electricity and magnetism were originally thought to be unrelated Electricity and magnetism were originally thought to be unrelated in 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena in 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena

Maxwell’s Starting Points Electric field lines originate on positive charges and terminate on negative charges Electric field lines originate on positive charges and terminate on negative charges Magnetic field lines always form closed loops – they do not begin or end anywhere Magnetic field lines always form closed loops – they do not begin or end anywhere A varying magnetic field induces an emf and hence an electric field (Faraday’s Law) A varying magnetic field induces an emf and hence an electric field (Faraday’s Law) Magnetic fields are generated by moving charges or currents (Ampère’s Law) Magnetic fields are generated by moving charges or currents (Ampère’s Law)

Maxwell’s Predictions Maxwell used these starting points and a corresponding mathematical framework to prove that electric and magnetic fields play symmetric roles in nature Maxwell used these starting points and a corresponding mathematical framework to prove that electric and magnetic fields play symmetric roles in nature He hypothesized that a changing electric field would produce a magnetic field He hypothesized that a changing electric field would produce a magnetic field Maxwell calculated the speed of light to be 3x10 8 m/s Maxwell calculated the speed of light to be 3x10 8 m/s He concluded that visible light and all other electromagnetic waves consist of fluctuating electric and magnetic fields, with each varying field inducing the other He concluded that visible light and all other electromagnetic waves consist of fluctuating electric and magnetic fields, with each varying field inducing the other

Hertz’s Confirmation of Maxwell’s Predictions Heinrich Hertz was the first to generate and detect electromagnetic waves in a laboratory setting Heinrich Hertz was the first to generate and detect electromagnetic waves in a laboratory setting

James Clerk Maxwell’s Equations (1867) Pre-Maxwell E-field comes out from p-charge and terminates at negative charge. B-field cannot do like E-field. No magnetic monopole! Faraday’s law Ampere’s law

The velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments, agrees so exactly with the velocity of light calculated from the optical experiments, that we can scarcely avoid the inference that light consists in the transverse undulation of same medium which is the cause of electric and magnetic Phenomena x 10 8 m/s

Hertz’s Experiment (1887)

Hertz’s Basic LC Circuit When the switch is closed, oscillations occur in the current and in the charge on the capacitor When the switch is closed, oscillations occur in the current and in the charge on the capacitor When the capacitor is fully charged, the total energy of the circuit is stored in the electric field of the capacitor When the capacitor is fully charged, the total energy of the circuit is stored in the electric field of the capacitor At this time, the current is zero and no energy is stored in the inductorAt this time, the current is zero and no energy is stored in the inductor

LC Circuit, cont As the capacitor discharges, the energy stored in the electric field decreases As the capacitor discharges, the energy stored in the electric field decreases At the same time, the current increases and the energy stored in the magnetic field increases At the same time, the current increases and the energy stored in the magnetic field increases When the capacitor is fully discharged, there is no energy stored in its electric field When the capacitor is fully discharged, there is no energy stored in its electric field The current is at a maximum and all the energy is stored in the magnetic field in the inductorThe current is at a maximum and all the energy is stored in the magnetic field in the inductor The process repeats in the opposite direction The process repeats in the opposite direction There is a continuous transfer of energy between the inductor and the capacitor There is a continuous transfer of energy between the inductor and the capacitor

Hertz’s Experimental Apparatus An induction coil is connected to two large spheres forming a capacitor An induction coil is connected to two large spheres forming a capacitor Oscillations are initiated by short voltage pulses Oscillations are initiated by short voltage pulses The inductor and capacitor form the transmitter The inductor and capacitor form the transmitter

Hertz’s Experiment Several meters away from the transmitter is the receiver Several meters away from the transmitter is the receiver This consisted of a single loop of wire connected to two spheresThis consisted of a single loop of wire connected to two spheres It had its own inductance and capacitanceIt had its own inductance and capacitance When the resonance frequencies of the transmitter and receiver matched, energy transfer occurred between them When the resonance frequencies of the transmitter and receiver matched, energy transfer occurred between them

Hertz’s Conclusions Hertz hypothesized the energy transfer was in the form of waves Hertz hypothesized the energy transfer was in the form of waves These are now known to be electromagnetic wavesThese are now known to be electromagnetic waves Hertz confirmed Maxwell’s theory by showing the waves existed and had all the properties of light waves Hertz confirmed Maxwell’s theory by showing the waves existed and had all the properties of light waves They had different frequencies and wavelengthsThey had different frequencies and wavelengths