Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a...
-
Upload
clinton-pope -
Category
Documents
-
view
225 -
download
2
Transcript of Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a...
![Page 1: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/1.jpg)
Chapter 30
Inductance
![Page 2: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/2.jpg)
Self Inductance
• When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this in turn induces an emf in that same coil.
• This induced emf opposes the change in flux.
![Page 3: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/3.jpg)
Some Terminology
• Use emf and current when they are caused by batteries or other sources
• Use induced emf and induced current when they are caused by changing magnetic fields
• When dealing with problems in electromagnetism, it is important to distinguish between the two situations
![Page 4: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/4.jpg)
Self Inductance cont.
• The magnetic flux B passing through the N turns
of a coil is proportional to the current I in the coil. Therefore, we define the self-inductance, L as:
![Page 5: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/5.jpg)
Self Inductance cont.
• The emf induced in a coil of self-inductance L can be obtained from Faraday’s law:
![Page 6: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/6.jpg)
Self-Inductance, Coil Example
• A current in the coil produces a magnetic field directed toward the left (a)
• If the current increases, the increasing flux creates an induced emf of the polarity shown (b)
• The polarity of the induced emf reverses if the current decreases (c)
![Page 7: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/7.jpg)
Example
• Calculate the value of the inductance for a tightly wrapped solenoid that has a length of 5.0 cm and a cross-sectional area of 0.30 cm2 if the number of turns of the coil is 100.
![Page 8: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/8.jpg)
Solution
• The inductance is given by the following:
![Page 9: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/9.jpg)
Solution cont.
• We are given the number of turns but not the current or the flux; therefore, we need to determine the flux through the coil.
• The magnetic field of a solenoid is given as:
![Page 10: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/10.jpg)
Solution cont.
• In the previous equation n represents the number of turns per unit length and l is the length of the solenoid.
• The flux is equal to the field times the cross-sectional area:
![Page 11: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/11.jpg)
Solution cont.
• If we now substitute this expression into our equation for the inductance it becomes:
![Page 12: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/12.jpg)
Solution cont.
• The inductance for the coil is then:
![Page 13: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/13.jpg)
Inductance Units
• The SI unit of inductance is the henry (H)
• Named for Joseph
Henry (pictured here)
AsV
1H1
![Page 14: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/14.jpg)
Energy Stored in an Inductor
• An inductor is a device that can be used to store energy.
• If an inductor is placed in a circuit and a continually changing voltage is applied then an induced emf will be created inside the inductor.
![Page 15: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/15.jpg)
Lenz’s Law
• According to Lenz’s law the polarity of the induced emf is opposite to that of the generator voltage.
• Therefore, the generator must perform work on the charges to push them through the inductor against the induced emf.
![Page 16: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/16.jpg)
Energy of an Inductor
• The work done to push a small amount of charge through the inductor can be expressed as:
![Page 17: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/17.jpg)
Energy of an Inductor
• If we remember that the current is the time rate of change of the charge then we can the following:
![Page 18: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/18.jpg)
Energy of an Inductor
• After integration we get a relationship for the work in terms of the current in the inductor.
![Page 19: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/19.jpg)
Energy of an Inductor
• By the work energy theorem we get:
![Page 20: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/20.jpg)
Energy in a Solenoid
• If our inductor is a long solenoid then we can express the energy stored as:
![Page 21: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/21.jpg)
Energy of a Solenoid
• The magnetic field for a solenoid is given as:
• Therefore, the energy stored by a current in a long solenoid is:
![Page 22: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/22.jpg)
Energy Density of a Solenoid
• The energy density of the solenoid can be defined as the energy stored per unit volume and has the following form:
![Page 23: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/23.jpg)
Example
• A coil of length 0.50-m and 200 turns of wire per meter is used to store enough energy to run your 100 W per channel stereo receiver at full volume for 20 minutes.
• How much current must be supplied to the coil to accomplish this if the coil has a diameter of 0.07 m?
![Page 24: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/24.jpg)
Solution
• The energy needed to run the receiver is equal to the product of the power consumed by the receiver and the time at which it is running.
• The energy required to run a 100 W two channel stereo receiver for 20 minutes is:
![Page 25: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/25.jpg)
Solution cont.
• The energy of a coil can be expressed as:
![Page 26: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/26.jpg)
Solution cont.
• The current is then:
![Page 27: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/27.jpg)
Example: The Coaxial Cable
• Calculate L for the cable• The total flux is
• Therefore, L is
• The total energy is
ln2 2
I Ibo o
B a
μ μ bB dA dr
πr π a
ln2I
B oμ bL
π a
221
ln2 4
II oμ b
U Lπ a
![Page 28: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/28.jpg)
Mutual Inductance
• If two coils are placed next to each other and one coil has a changing current passing through it the second coil will experience a changing magnetic flux through it.
• This changing flux will produce an induced emf in the second coil.
![Page 29: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/29.jpg)
Mutual Inductance, 2
• The current in coil 1 sets up a magnetic field
• Some of the magnetic field lines pass through coil 2
• Coil 1 has a current I1
and N1 turns• Coil 2 has N2 turns
![Page 30: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/30.jpg)
Mutual Inductance cont.
• The relationship for mutual inductance is as follows:
![Page 31: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/31.jpg)
Mutual Inductance cont.
• In the previous equation M represents the mutual inductance, N2 is the number of
turns in the second coil, 2 is the flux
passing through the second coil, and I1 is
the current in the first coil.
• The units for inductance are called the Henry H.
![Page 32: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/32.jpg)
Mutual Inductance cont.
• If we apply Faraday’s law we can derive a relationship between the mutual inductance and the emf created in the second coil by the first.
![Page 33: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/33.jpg)
Example
• A long thin solenoid of length L and cross-sectional area A contains N1 closely packed turns of wire.
• Wrapped around it is an insulated coil of N2 turns.
• Assume all the flux from coil 1 (the solenoid) passes through coil 2, and calculate the mutual inductance.
![Page 34: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/34.jpg)
Solution
• We first determine the flux produced by the solenoid.
• The magnetic field inside the solenoid is given by the following:
![Page 35: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/35.jpg)
Solution cont.
• The flux induced in the second coil due to the first is then:
![Page 36: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/36.jpg)
Solution cont.
• Hence the mutual inductance is:
![Page 37: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/37.jpg)
The RLC Circuit
• Consider a circuit with an inductor, a charged capacitor, and a resistor all hooked together in series.
• The energy stored in the capacitor will begin to flow over into the inductor.
• Meanwhile, the current in the circuit will be dissipated by the resistor.
![Page 38: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/38.jpg)
The RCL Circuit cont.
• The equation describing this can be written as:
![Page 39: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/39.jpg)
The RCL Circuit cont.
• If we now divide the equation by the current and then use the relation that the current is the time derivative of the charge then we get the following:
![Page 40: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/40.jpg)
The RCL Circuit cont.
• This is an ordinary second order differential equation with constant coefficients.
• It has solutions that look like the following:
![Page 41: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/41.jpg)
The RCL Circuit cont.
• The angular frequency in the previous equation is the circuit’s damped oscillating frequency and is given by:
![Page 42: Chapter 30 Inductance. Self Inductance When a time dependent current passes through a coil, a changing magnetic flux is produced inside the coil and this.](https://reader034.fdocuments.in/reader034/viewer/2022042717/56649e0f5503460f94af9e61/html5/thumbnails/42.jpg)
The RCL Circuit cont.
• Note: in the absence of a resistor, the solution to the differential equation is that of a harmonic oscillator with an angular frequency of: