Lecture 12. Energy, Force, and Work in Electro- and Magneto-Quasistatics

Transcription

1 Lecture 1 Energy, Force, an ork in Electro an MagnetoQuasistatics n this lecture you will learn: Relationship between energy, force, an work in electroquasistatic an magnetoquasistatic systems ECE 303 Fall 007 Farhan Rana Cornell University Lorentz Electric Force The force on a charge q in an Efiel is: r r F qe q E Eample: consier two charges q 1 an q : nˆ q 1 q 1 The force that q 1 eerts upon q can be obtaine by multiplying the Efiel that q 1 prouces at the location of q by the charge q : r F 1 r q E q 1 q 4π ε nˆ q q o 4π ε o nˆ 1 The force that q eerts upon q 1 is just equal an opposite to what q 1 eerts upon q (Newton s thir law): r q1 q q1 q F 1 nˆ 1 nˆ 1 4π εo 4π ε o ECE 303 Fall 007 Farhan Rana Cornell University 1

2 Force Between Charge Plates Consier two charge plates carrying charge per unit areas of an, respectively Nee to fin the total force on the right plate eerte by the left plate (which is also equal an opposite to the force on the left plate eerte by the right plate) area A 0 Step (1): Fin the Efiel at the location of the right plate prouce by the left plate ASSUMNG THE RGHT PLATE AS NOT THERE: E ( ) no right plate ε Step (): Multiply the Efiel calculate above by the total charge on the right plate to get the esire force F L R ( A ) E ( ) ECE 303 Fall 007 Farhan Rana Cornell University o no right plate A ε o Potential Energy an Force The force on a charge q in an Efiel is: E F qe q The potential energy of a charge in the Efiel is: U q φ ( ) qe So the force on the charge can also be written as: F U Electric force is the erivative of potential energy This formula is much more general than it appears (see the net few slies..) ECE 303 Fall 007 Farhan Rana Cornell University

3 Energy, Force, an ork in Electromagnetics Lets generalize the relationship between energy an force Suppose one has a CLOSED electromagnetic system with electromagnetic energy U f the system oes some mechanical work F then its electromagnetic energy must ecrease by the same amount: U F or U F Here F is some force that results in the change of some system length parameter by ECE 303 Fall 007 Farhan Rana Cornell University Now lets use the energy concepts to calculate the same force Total electric energy store in the fiel can be calculate as follows: r r ε E E U o. εo ( A ) εo Force Between Charge Plates The force between the plates can move the plates thereby oing work that woul result in a change of the value of F εo A A o ε ε o area A The negative sign inicates the force is in the irection of ecreasing i.e. the force between the plates is attractive 0 ECE 303 Fall 007 Farhan Rana Cornell University 3

4 Force Between Charge Metal Plates area A Now consier the force between the charge metal plates of a parallel plate capacitor connecte to a voltage source hat is the force between the plates in terms of the applie voltage? Answer shoul be the same as before: A F ε o 1 C φ φ 0 But now when you use the previous formula you will get the wrong sign: 1 C F hat went wrong? 1 C ( ) E Q C C ε o A 1 U C ε o The presence of a voltage source means you on t have a close system anymore ECE 303 Fall 007 Farhan Rana Cornell University area A Force Between Charge Metal Plates φ φ 0 f the system tries to o work by bringing the plates closer or pushing them further apart, the voltage source ensures that the charge Q in the capacitor always satisfies the relation: QC ( ) E Q C C ε o A 1 U C ε o The voltage source ensures that Q is always equal to C by bringing in or removing charge from the capacitor while the mechanical work is being performe This work one by the voltage source in bringing in or removing charges must also be inclue in the analysis ECE 303 Fall 007 Farhan Rana Cornell University 4

5 Energy, Force, an ork in Electromagnetics oltage Sources Lets further generalize the relationship between energy an force by incluing voltage sources Suppose one has an electromagnetic system with electromagnetic energy U that is connecte to a FXED voltage source f the system performs mechanical work F then its electromagnetic energy U must ecrease by the same amount f the voltage source passes a current in time t then it oes work an the system electromagnetic energy U must also increase by the same amount So instea of: e write: U F U F t But if Q is the total charge that passe in time t then: U F Q Q t F Assuming voltage is hel fie ECE 303 Fall 007 Farhan Rana Cornell University Energy, Force, an ork in Electromagnetics oltage Sources More generally the total ifferential of energy U can be written as: U But we ha: Q fie U F Q fie Q Q Therefore, it must be that: F an Q fie Q fie But if the thing that is kept fie is the voltage then we alreay have the result: F fie ECE 303 Fall 007 Farhan Rana Cornell University 5

6 Electric energy is: 1 U C F 1 C 1 C Force Between Charge Metal Plates fie C area A Since the voltage is hel fie the force between the plates can be calculate as: φ φ 0 E ( ) Q C C A ε o 1 U C ε o This time we have got the sign right as well ECE 303 Fall 007 Farhan Rana Cornell University Application: An Electrostatic Actuator The applie voltage can be use to pull in the Depth of plates t ielectric slab an when the voltage is remove Area of plates Lt the slab will come own by gravity φ φ 0 Electric energy is: L 1 U C y Force on the slab is: F y y 1 C y 1 ( ε ε ) t o fie C y ve sign of the force means that the force is in the irection of increasing y, i.e. the slab will be pulle in when a voltage is applie ECE 303 Fall 007 Farhan Rana Cornell University ε E ε ( L y ) t Q C o ε yt C 1 U C 6

7 Force on a Metal ire Over a Metal Groun Plane Consier a very long (in the zirection) metal wire of raius a over a metal groun plane an at a istance / from it Nee to fin the force per unit length between the metal wire an the groun plane when the potential ifference between them is volts Electric energy per unit length is: y raius a 1 U C Force per unit length on the wire is: F 1 C 1 C fie C 1 C ln a ECE 303 Fall 007 Farhan Rana Cornell University From homework # Q C πε o C ln a 1 U C Lorentz Magnetic Force The force on a charge q moving in a magnetic fiel is: r r r F q v B q B v r The force per unit length on a wire carrying current is: r r r F q v B. number of charges per unit length r But: { } { } r r r r F B { q v}. { number of charges per unit length} r B ECE 303 Fall 007 Farhan Rana Cornell University 7

8 Electric Motors A coil of a current carrying wire in a magnetic fiel will eperience a torque as its two ens will eperience forces in opposite irections F r L r r r F ( B )L F r B ECE 303 Fall 007 Farhan Rana Cornell University H K µ H L o l µ o l (L is the total inuctance) Energy, Forces, an ork in Magnetics y The structure has length l in the zirection Recall that: K K zˆ H K zˆ Question: hat is the force between the two metal plates? s it repulsive or is it attractive? Try the same proceure: if a system oes some mechanical work F then its electromagnetic energy must ecrease by the same amount: U U F or F Magnetic systems that require current from outsie sources are NOT REALLY CLOSED SYSTEMS ECE 303 Fall 007 Farhan Rana Cornell University 8

9 Energy, Forces, an ork in Electromagnetics Current Sources Suppose one has an electromagnetic system with electromagnetic energy U that is connecte to a FXED current source f the system performs mechanical work F then its electromagnetic energy U must ecrease by the same amount f the current source passes current uner a potential in time t then it oes work an the system electromagnetic energy U must also increase by the same amount So instea of: e write: U F U F λ F t But if the electromagnetic system is an inuctor then we know that: U F λ Assuming current is hel fie λ t The prouct λ is the work one by the current source in changing the flu by λ ECE 303 Fall 007 Farhan Rana Cornell University Energy, Forces, an ork in Electromagnetics Current Sources More generally the total ifferential of energy U can be written as: U λ fie But we ha: λ U F λ Therefore, it must be that: F fie λ an λ fie λ fie But if the thing that is kept fie is the current then we alreay have the result: λ F fie ECE 303 Fall 007 Farhan Rana Cornell University 9

10 Force Between Parallel Current Carrying Plates y Force between the plates in a parallel plate inuctor The structure has length l in the zirection K zˆ H K H λ L L µ o l 1 U L K K zˆ λ F fie 1 L 1 L F The ve sign inicates that the force is in the irection of increasing Force is repulsive the plates repel each other ECE 303 Fall 007 Farhan Rana Cornell University Force Between a Metal ire an a Metal Plane y Force between a current carrying wire an a perfect metal plane The structure has length l in the zirection raius a zˆ λ L L µ o l log π a 1 U L λ F fie 1 L 1 L F log a The ve sign inicates that the force is in the irection of increasing Force is repulsive the metal wire is repelle by the metal plane ECE 303 Fall 007 Farhan Rana Cornell University 10

11 Application: Magnetic Levitation The inuctance L of the coil is an increasing function of the istance from the metal block (. because of image currents) λ L L L( ) 1 U L λ F fie 1 L F The ve sign inicates that the force is in the irection of increasing Force is repulsive the metal coil is repelle by the metal plane ECE 303 Fall 007 Farhan Rana Cornell University ECE 303 Fall 007 Farhan Rana Cornell University 11