Validity of expressions

Transcription

1 E&M Lecture 8 Topics: (1)Validity of expressions ()Electrostatic energy (in a capacitor?) (3)Collection of point charges (4)Continuous charge distribution (5)Energy in terms of electric field (6)Energy stored in capacitor (7)Energy density

2 Validity of expressions Always alid: Gauss Law for E, P and D relation D = ε o E + P Limited alidity: expressions inoling ε r and χ e Hae assumed that χ e is a simple number: P = χ e ε o E Which is only true in LIH media: Linear: χ e independent of magnitude of E; interesting media non-linear : P = χ 1 ε o E + χ ε o E +. Isotropic: χ e independent of direction of E; interesting media anisotropic : χ e is a tensor (gen. ector) Homogeneous: uniform medium (recall arying ε r and ρ b )

3 Electrostatic Energy recall energy stored in a capacitor: U = QV/ Raises questions: (1) why factor of 1/? () where exactly is the energy stored? Re (1) energy is charge times potential difference But in this case, the charge stored is creating the potential difference: the factor of 1/ is to aoid double-counting! Or graph the charging process.. V can further this discussion by collecting point charges.. Also, begin discussion, as always, in a acuum. Q

4 Collection of Point Charges in acuum potential is energy per unit +e charge Consider isolated point charge q 1 Bringing a second point charge q from infinity to reside at r 1 from q 1 Potential energy is q q 1 4πε o r 1 q 1 q r 1 Now bring a third point charge q 3 from infinity to reside at r 13 from q 1 and at r 3 from q Additional potential energy is q q πε o r 13 arriing charge times existing potential q 4πε o r 3

5 Energy of Collection of Point Charges Extending the process, the total potential energy is q U = q 1 q + q πε o r 1 4πε o r 13 Each bracket contains all point charges which pre-existed the one outside! Re-write as: U = 1 4πε o Note how the condition j < i preents double counting! Exploit this by deliberately double-counting and diiding by two - or introducing a factor of 1/! q 4πε o r 3 i q i j<i + q 4 q j r ji...

6 Point charges s continuous distribution U = 1 1 4πε o q i i Where φ i is the potential due to all point charges except q i Note (a) for single point charge, U = 0 (b) U can be +e or -e (key assumption: already-assembled point charges) Extend to continuous distribution: replace q by dq = ρd Where is any olume enclosing all the charge q j = 1 q φ i i j i r ji This expression includes energy of assembly and is always +e! Show this and obtain useful expression.. i U = 1 φρd

7 Energy in acuum in terms of E.E = ρ ε o and E = φ φ = ρ ε o ρ = ε o φ U = 1 φρd = ε o Where φ is sole parameter; expand integrand using identity:.ψf = ψ.f + F. ψ Exercise: write ψ = φ and F = φ to show: φ φd.φ φ = φ φ + ( φ) substitute.. φ φ =.φ φ ( φ)

8 U = ε o = ε o Energy in acuum in terms of E.φ φd φ ( φ φ).da φ s ( ) d ( ) d Recall where is any olume enclosing all the charge Free choice of s, the enclosing surface - so let it expand! Think of sphere, radius r : da goes as r but φ goes as r -1 (at least) and φ goes as r - (at least) Hence surface integral goes as r -1 (at least) For large r, this integral goes to zero, the olume integral remains unchanged! Green' s Theorem ( )

9 Energy in acuum in terms of E U = ε o ( φ φ).da φ s ( ) d = + ε o ( φ) d = ε o E d where encloses all regions where E is non-zero. Confirms mathematically that U is always +e: E-squared Only applies to acuum!!! Use this expression to calculate electrostatic energy..example of (acuum) capacitor

10 Energy stored in (acuum) capacitor U = ε o = ε o V d E d d = ε o E d = ε o As E is constant and equal to V/d, and the olume is A x d this example is triial, but illustrates the answer to the second question raised in the introduction: where in the capacitor does the energy reside? - it resides in the field! V d A d = 1 ε o A d V = 1 CV = 1 QV

11 Energy density in a acuum U = ε o E d du d = ε o E This is the energy density and as field E is a point concept, the energy density is also a point concept. In next lecture, we determine this expression for the energy density ia a less mathematical and more physical argument!