Phys2212_L06 - Physics 2212 Electricity and Magnetism...

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Physics 2212 Electricity and Magnetism Lecture 6 Flux and Gauss’s Law
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10/06/09 Physics 2212 - Lecture 6 2 Chapter 26 Summary (1)
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10/06/09 Physics 2212 - Lecture 6 3 Chapter 26 Summary (2)
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10/06/09 Physics 2212 - Lecture 6 4 Types of Symmetry (again) Full rotational symmetry: any rotation about any axis does not change object: Sphere Cylindrical symmetry: any rotation about one axis does not change object: Cylinder Partial rotational symmetry: 72 0 rotation about one axis does not change object: Star 72 0 Reflectional symmetry: mirror image reflection about some plane is same as object. A (yes) A E (no) E Translational symmetry: translation along a line does not change object. Symmetry Principle: The electric field must have the same symmetries as the charge distribution that produced it.
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10/06/09 Physics 2212 - Lecture 6 5 Symmetry of a Long Cylinder A long cylinder has the following symmetries: 1. Translational symmetry along the z axis; 2. Full rotational symmetry about the z axis; 3. z axis; 4. Reflectional symmetry in any plane the z axis. z
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10/06/09 Physics 2212 - Lecture 6 6 Rejected Possibilities (1) Question: Could the E field have z-axis components? Symmetry Answer: Consider the 4 th symmetry: (reflection in a plane z) The field changes on this reflection, but the charged object does not. Therefore, the E-field of the long cylinder cannot have z-axis components
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10/06/09 Physics 2212 - Lecture 6 7 Rejected Possibilities (2) Question: Could the E-field have θ -axis components? Symmetry Answer: Consider the 3rd symmetry: (reflection plane contains z) The field changes on this reflection, but the charged object does not. Therefore, the E-field cannot have θ -axis components. Conclusion: In cylindrical coordinates, the electric field must be radial (r), with no components in the θ or z directions . z r r θ
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10/06/09 Physics 2212 - Lecture 6 8 What Good is Symmetry? Symmetry allows us to rule out many conceivable field shapes that are found to be incompatible with the symmetries of the charge distribution. Knowing what cannot happen is often as useful as knowing what must happen. By using symmetry arguments, we can avoid lengthy mathematical calculations that will ultimately give a zero result.
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Phys2212_L06 - Physics 2212 Electricity and Magnetism...

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