Electrostatics: Energy in Electrostatic Fields

Transcription

1 7/4/08 Electrostatics: Energy in Electrostatic Fields EE33 Electromagnetic Field Theory Outline Energy in terms of potential Energy in terms of the field Power and energy in conductors Electrostatics -- Energy Slide

2 7/4/08 Energy in Terms of Potential Recall Potential Difference A d B Recall the relation between potential difference, work, and charge. W V V V Q E AB B A Therefore, the work it takes to moe charge Q from A to B is W QV AB Electrostatics -- Energy Slide 4

3 7/4/08 Energy in an Ensemble of Charges Q Q Q 3 An ensemble of charges contains energy because the charges are putting a force on each other and so they hae the potential to do work. We will calculate how much energy the ensemble contains by calculating how much energy it took to assemble it. Electrostatics -- Energy Slide 5 Point Charge # Q P No other charges are present, so placing Q at P takes no work. W 0 Electrostatics -- Energy Slide 6 3

4 7/4/08 Point Charge # Q Q P Placing Q at P takes work because charge Q is present. W QV Electrostatics -- Energy Slide 7 Point Charge #3 Q Q P Q3 3 Placing Q 3 at P 3 takes work because charges Q and Q are present. W QV QV Electrostatics -- Energy Slide 8 4

5 7/4/08 Total Work So Far Q Q Q 3 The total work placing all three charges is W WW W3 0 QV Q V V Electrostatics -- Energy Slide 9 Assembly in Reerse Order Q Q Q 3 If we had placed the charges in the reerse order, W W3W W 0 QV Q V V 3 3 Electrostatics -- Energy Slide 0 5

6 7/4/08 Add Both Approaches W 0 QV Q V V W 0 QV Q V V 3 3 Equation obtained by placing Q, then Q, and then Q 3. Equation obtained by placing Q 3, then Q, and then Q QV 3 Q V V3 W 0QV Q V V 0 Add the two equations aboe. W Q V V Q V V Q V V V V V3 W QV QV QV 3 3 Electrostatics -- Energy Slide Final Expression W QV QV QV 3 3 W QV QV QV Sole for W. 3 3 It is straightforward to generalize this for any number of charges. W N i QV i i joules Electrostatics -- Energy Slide 6

7 7/4/08 Energy in Charge Distributions Point Charge Line Charge Sheet Charge Volume Charge Q s Charge Q C Line Charge Density C m Surface Charge Density C m s Volume Charge Density C m 3 Total Charge Q Total N Qi i Total Charge Q d L Total Total Charge Q ds S Total s s S Total Charge Q d V Total Total Energy N W i QV i i Total Energy W Vd L Total Energy W svds s Total Energy W Vd Electrostatics -- Energy Slide 3 Energy in Terms of the Field 7

8 7/4/08 Deriation ( of 5) The energy in a olume charge is W Vd Recall from Maxwell s equations that D. W DVd Recall the product rule for diergence fa f A Af VD V DDV D V VD DV Electrostatics -- Energy Slide 5 Deriation ( of 5) Apply the product rule for our equation for work. W DVd VDD V d VDd D V d Electrostatics -- Energy Slide 6 8

9 7/4/08 Deriation (3 of 5) Recall the diergence theorem S Fds F d W VDd DVd Apply this to our equation for work. V VD ds S W VDds DVd S Electrostatics -- Energy Slide 7 Deriation (4 of 5) Look more closely at the surface integral. W VDds DVd V r S D r ds r Oerall r r r r We are free to choose whateer surface S we wish. As we enlarge the surface out to infinity, the surface integral becomes negligible relatie to the olume integral. W VDds DVd S Electrostatics -- Energy Slide 8 9

10 7/4/08 Deriation (5 of 5) Our equation for work is now W DVd Associate the negatie sign with V. W D V d This is the electric field intensity. W D E d This is the general equation for energy stored in the electrostatic field. It is alid for anisotropic and inhomogeneous media. E Electrostatics -- Energy Slide 9 Electrostatic Energy in LHI Media The more common expression for energy in the electrostatic field is for the special case of linear, homogeneous, and isotropic (LHI) media. In isotropic media we hae D E. W DEd E E d W E d Simpler equation that is only alid in LHI media. Electrostatics -- Energy Slide 0 0

11 7/4/08 Electrostatic Energy Density Obsere what we hae been integrating to get total energy. W DEd These expressions must be energy density w. W E d We can now think of calculating total energy by integrating the energy density w. D E General case W wd w E LHI media Electrostatics -- Energy Slide Power & Energy in Conductors

12 7/4/08 Joule s Law Joule s law states that This is equialent to P = VI in circuit theory. P E J d From this, we can extract the energy density in a conductor. w EJ Applying Ohm s law for electromagnetics J E gies w EJ E E P E d Most common form. E Electrostatics -- Energy Slide 3