gauss’ law (chapter 28) - sfu.ca · 2010-03-01 · gauss’ law (chapter 28) to figure out the...
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![Page 1: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/1.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.
Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.Which one should you use?In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 2: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/2.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.
If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.Which one should you use?In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 3: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/3.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.
Which one should you use?In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 4: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/4.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.Which one should you use?
In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 5: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/5.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.Which one should you use?In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).
Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 6: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/6.jpg)
Gauss’ Law (Chapter 28)
To figure out the force on a particle due to a charge (or to definethe field) we used Coloumb’s Law.Coloumb’s Law is very useful in many circumstances, butsometimes another approach would be useful - that is whereGauss’ Law comes in.If you recall your mechanics experience, sometimes you usedNewton’s Laws to solve things, sometimes the conservation ofmomentum or energy.Which one should you use?In principle it doesn’t matter...they are equivalent:
�F = m�a =d�pdt
Newton’s Laws can be derived from Conservation of Momentum(and vice versa).Coloumb’s Law and Gauss’ Law can be derived from each othertoo. Use whichever is easiest/best for a certain situation.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 1 / 8
![Page 7: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/7.jpg)
Symmetry (28.1)
SYMMETRY — YRTEMMYS
Symmetries are veryimportant in physics.Exploiting symmetries willmake solving problems mucheasier.What is a symmetry??
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 2 / 8
![Page 8: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/8.jpg)
Symmetry (28.1)
SYMMETRY — YRTEMMYSSymmetries are veryimportant in physics.
Exploiting symmetries willmake solving problems mucheasier.What is a symmetry??
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 2 / 8
![Page 9: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/9.jpg)
Symmetry (28.1)
SYMMETRY — YRTEMMYSSymmetries are veryimportant in physics.Exploiting symmetries willmake solving problems mucheasier.
What is a symmetry??
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 2 / 8
![Page 10: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/10.jpg)
Symmetry (28.1)
SYMMETRY — YRTEMMYSSymmetries are veryimportant in physics.Exploiting symmetries willmake solving problems mucheasier.What is a symmetry??
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 2 / 8
![Page 11: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/11.jpg)
What is Symmetry
SymmetryA symmetry is an operation you can perform on a system which leavesthe system unchanged. In other words, you cannot tell that you didanything!
Consider an equilateral triangle as an example
You can rotate this by 120 degrees and not know that you did it. Youcan flip it about axis Aa (or similar axes) and not know that you did it.These are symmetry operations.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 3 / 8
![Page 12: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/12.jpg)
What is Symmetry
SymmetryA symmetry is an operation you can perform on a system which leavesthe system unchanged. In other words, you cannot tell that you didanything!
Consider an equilateral triangle as an example
You can rotate this by 120 degrees and not know that you did it.
Youcan flip it about axis Aa (or similar axes) and not know that you did it.These are symmetry operations.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 3 / 8
![Page 13: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/13.jpg)
What is Symmetry
SymmetryA symmetry is an operation you can perform on a system which leavesthe system unchanged. In other words, you cannot tell that you didanything!
Consider an equilateral triangle as an example
You can rotate this by 120 degrees and not know that you did it. Youcan flip it about axis Aa (or similar axes) and not know that you did it.These are symmetry operations.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 3 / 8
![Page 14: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/14.jpg)
Symmetry
Imagine an infinitely long cylinder fullof charges. What geometricoperations could you perform on thatcylinder and not know that you didit??
Well, we could
Translate the charge along thecylinderRotate the charge about an axisReflect the charge in a mirror
(See the figures to the left)
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 8
![Page 15: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/15.jpg)
Symmetry
Imagine an infinitely long cylinder fullof charges. What geometricoperations could you perform on thatcylinder and not know that you didit??Well, we could
Translate the charge along thecylinderRotate the charge about an axisReflect the charge in a mirror
(See the figures to the left)
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 8
![Page 16: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/16.jpg)
Symmetry
Imagine an infinitely long cylinder fullof charges. What geometricoperations could you perform on thatcylinder and not know that you didit??Well, we could
Translate the charge along thecylinder
Rotate the charge about an axisReflect the charge in a mirror
(See the figures to the left)
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 8
![Page 17: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/17.jpg)
Symmetry
Imagine an infinitely long cylinder fullof charges. What geometricoperations could you perform on thatcylinder and not know that you didit??Well, we could
Translate the charge along thecylinderRotate the charge about an axis
Reflect the charge in a mirror
(See the figures to the left)
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 8
![Page 18: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/18.jpg)
Symmetry
Imagine an infinitely long cylinder fullof charges. What geometricoperations could you perform on thatcylinder and not know that you didit??Well, we could
Translate the charge along thecylinderRotate the charge about an axisReflect the charge in a mirror
(See the figures to the left)
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 4 / 8
![Page 19: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/19.jpg)
Symmetry
The field on the left does not remain the same under a reflection!
That is not going to work because the charge distribution whichgenerates the field is symmetric with respect to reflection.
The symmetry of the electric field must match the symmetry of thecharge distribution.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 5 / 8
![Page 20: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/20.jpg)
Symmetry
The field on the left does not remain the same under a reflection!That is not going to work because the charge distribution whichgenerates the field is symmetric with respect to reflection.
The symmetry of the electric field must match the symmetry of thecharge distribution.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 5 / 8
![Page 21: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/21.jpg)
Symmetry
The field on the left does not remain the same under a reflection!That is not going to work because the charge distribution whichgenerates the field is symmetric with respect to reflection.
The symmetry of the electric field must match the symmetry of thecharge distribution.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 5 / 8
![Page 22: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/22.jpg)
Symmetry
The field on the left does not remain the same under a reflection!That is not going to work because the charge distribution whichgenerates the field is symmetric with respect to reflection.
The symmetry of the electric field must match the symmetry of thecharge distribution.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 5 / 8
![Page 23: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/23.jpg)
Symmetry
The previous page showed us that anelectric field component parallel to thecylindrical axis will not work.
However, what about the electric fieldpictured on the left?Nope. Also bad. If we reflect itthrough a plane containing the axis Ican tell it changed!
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 6 / 8
![Page 24: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/24.jpg)
Symmetry
The previous page showed us that anelectric field component parallel to thecylindrical axis will not work.However, what about the electric fieldpictured on the left?
Nope. Also bad. If we reflect itthrough a plane containing the axis Ican tell it changed!
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 6 / 8
![Page 25: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/25.jpg)
Symmetry
The previous page showed us that anelectric field component parallel to thecylindrical axis will not work.However, what about the electric fieldpictured on the left?Nope. Also bad. If we reflect itthrough a plane containing the axis Ican tell it changed!
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 6 / 8
![Page 26: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/26.jpg)
Symmetry
Well, there is only one shape left!
The electric field must be radial, pointing straight out from thecenter of the cylinder.You can see already how we exploit symmetries. We have usedthem to rule out possible field shapes.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 7 / 8
![Page 27: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/27.jpg)
Symmetry
Well, there is only one shape left!The electric field must be radial, pointing straight out from thecenter of the cylinder.
You can see already how we exploit symmetries. We have usedthem to rule out possible field shapes.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 7 / 8
![Page 28: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/28.jpg)
Symmetry
Well, there is only one shape left!The electric field must be radial, pointing straight out from thecenter of the cylinder.You can see already how we exploit symmetries. We have usedthem to rule out possible field shapes.
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 7 / 8
![Page 29: Gauss’ Law (Chapter 28) - SFU.ca · 2010-03-01 · Gauss’ Law (Chapter 28) To figure out the force on a particle due to a charge (or to define the field) we used Coloumb’s](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f105d607e708231d448bf66/html5/thumbnails/29.jpg)
Some Fundamental Symmetries
Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring 2010 8 / 8