组合数学 (Spring 2014)/Problem Set 3 and Permeability (electromagnetism): Difference between pages

Magnetic Circuits
	Conventional Magnetic Circuits Magnetomotive force [math]\displaystyle{ \mathcal F }[/math]; Magnetic flux [math]\displaystyle{ \Phi }[/math]; Magnetic reluctance [math]\displaystyle{ \mathcal R }[/math];
	Phasor Magnetic Circuits Complex reluctance [math]\displaystyle{ Z_\mu }[/math];
	Related Concepts Magnetic permeability [math]\displaystyle{ \mu }[/math];
	Gyrator-capacitor model variables Magnetic impedance [math]\displaystyle{ z_M }[/math]; Effective resistance [math]\displaystyle{ r_M }[/math]; Magnetic inductivity [math]\displaystyle{ L_M }[/math]; Magnetic capacitivity [math]\displaystyle{ C_M }[/math];
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Latest revision as of 16:23, 28 June 2016

Permeability is a property of a material that describes how dense a magnetic field would be if the same amount of current was passed through it. Permeability is measured in henries per metre (H/m) and its symbol is [math]\displaystyle{ \mu }[/math].

Since empty space has a constant permeability (called the permeability of free space or [math]\displaystyle{ \mu_{0} }[/math]) of exactly [math]\displaystyle{ 0.0000004 \times \pi }[/math], most materials are listed with a relative permeability (symbol [math]\displaystyle{ \mu_{r} }[/math]). Relative permeability is the permeability of the material divided by the permeability of free space ([math]\displaystyle{ \mu_{r} = \mu / \mu_{0} }[/math]). The permeability of most materials is very close to 1. That means that the permeability of most materials is close enough that we can typically ignore it and use the permeability of free space instead.^[1] The biggest exceptions are materials called ferromagnetic materials. Some examples are iron (5000) and nickel (600). Some materials have been specially designed to have a permeability one million times larger than empty space.^[2]

References

Template:Reflist

↑ Lines and Fields in Electronic Technology, Stanley and Harrington pg 13
↑ http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur

[1] Lines and Fields in Electronic Technology, Stanley and Harrington pg 13

[2] ttp://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur

[1]

[2]

@@ Line 1: / Line 1: @@
-==Problem 1 ==
+{{MagneticCircuitSegments}}
-Recall that <math>\chi(G)</math> is the chromatic number of graph <math>G</math>.
+'''Permeability''' is a [[property]] of a material that describes how [[density|dense]] a [[magnetism|magnetic field]] would be if the same amount of [[electric current|current]] was passed through it. Permeability is measured in henries per [[metre]] (H/m) and its symbol is <math>\mu</math>.
-Prove:
+Since [[vacuum|empty space]] has a [[constant]] permeability (called the '''permeability of free space''' or <math>\mu_{0}</math>) of exactly <math>0.0000004 \times \pi</math>, most materials are listed with a ''relative permeability'' (symbol <math>\mu_{r}</math>). Relative permeability is the permeability of the material divided by the permeability of free space (<math>\mu_{r} = \mu / \mu_{0}</math>). The permeability of most materials is very close to 1. That means that the permeability of most materials is close enough that we can typically ignore it and use the permeability of free space instead.<ref>Lines and Fields in Electronic Technology, Stanley and Harrington pg 13</ref> The biggest exceptions are materials called [[ferromagnetism|ferromagnetic materials]]. Some examples are [[iron]] (5000) and [[nickel]] (600). Some materials have been specially designed to have a permeability one million times larger than empty space.<ref>http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur</ref>
-* Any graph <math>G</math> must have at least <math>{\chi(G)\choose 2}</math> edges.
-* For any two graphs <math>G(V,E)</math> and <math>H(V,F)</math>. Prove that <math>\chi(G\cup H)\le\chi(G)\chi(H)</math>.
-==Problem 2 ==
+== References ==
-(Erdős-Lovász 1975)
+{{reflist}}
-Let <math>\mathcal{H}\subseteq{V\choose k}</math> be a <math>k</math>-uniform <math>k</math>-regular hypergraph, so that for each <math>v\in V</math> there are ''exact'' <math>k</math> many <math>S\in\mathcal{H}</math> having <math>v\in S</math>.
+[[Category:Electromagnetism]]
-Use the probabilistic method to prove: For <math>k\ge 10</math>, there is a two coloring <math>f:V\rightarrow\{0,1\}</math> such that <math>\mathcal{H}</math> does not contain any monochromatic hyperedge <math>S\in\mathcal{H}</math>.
-== Problem 3 ==
-(Frankl 1986)
-Let <math>\mathcal{F}\subseteq {[n]\choose k}</math> be a <math>k</math>-uniform family, and suppose that it satisfies that <math>\A\cap B \not\subset C</math> for any <math>A,B,C\in\mathcal{F}</math>.
-* Fix any <math>B\in\mathcal{F}</math>. Show that the family <math>\{A\cap B\mid A\in\mathcal{F}, A\neq B\}</math> is an anti chain.
-* Show that <math>|\mathcal{F}|\le 1+{k\choose \lfloor k/2\rfloor}</math>.
-== Problem 4 ==
-Given a graph <math>G(V,E)</math>, a ''matching'' is a subset <math>M\subseteq E</math> of edges such that there are no two edges in <math>M</math> sharing a vertex, and a ''star'' is a subset <math>S\subseteq E</math> of edges such that every pair <math>e_1,e_2\in S</math> of distinct edges in <math>S</math> share the same vertex <math>v</math>.
-Prove that any graph <math>G</math> containing more than <math>2(k-1)^2</math> edges either contains a matching of size <math>k</math> or a star of size <math>k</math>.

组合数学 (Spring 2014)/Problem Set 3 and Permeability (electromagnetism): Difference between pages

Latest revision as of 16:23, 28 June 2016

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