elec%4705%lec%2% - carleton universitytjs/classicalphysics.pdf · c2 @ e @ t (1) from the math we...
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Elec%4705%Lec%2%
Classical%Physics%
Operators%and%differen;al%Eq’s%• The%most%sophis;cated%and%powerful%method%of%
expressing%classical%physics%is%through%the%development%of%differen;al%equa;ons.%%
• OGen%these%are%par;al%differen;al%equa;ons%which%are%func;ons%of%;me%(t)%and%space%r%=%{x,y,z}.%%
• We%do%this%through%operators.%%• We%replace%the%simple%math%and%laws%such%as%F%=%ma%
with%more%powerful%expressions%QQ%very%oGen%differen;al%operators.%
Operators%and%differen;al%Eq’s%• We%will%now%define%some%of%these%operators.%%• Operators%are%mathema;cal%tools%that%“operate”%QQ%in%other%words%perform%a%manipula;on%on%a%func;on.%%
• An%example%is%integra;on%or%differen;a;on.%%• Using%operators%we%can%reformulate%the%laws%of%physics%in%powerful%concise%expressions%that%allow%us%to%solve%difficult%problems%QQ%some;mes%with%difficulty!%
• Our%primary%operator%is%the%“del”%operator%�%
The%``Del''%operator%QQ%%%• Vector%operator!%• A%summary%of%some%mathema;cal%opera;ons%
rr = i
@
@x+ j
@
@y+ k
@
@z
operation name result
rV Gradient Vector
r . V Divergence Scaler
r2V Laplacian Scaler
r⇥ V Curl Vector
Gradient%• The%gradient%can%be%thought%of%as%essen;ally%the%slope%of%a%func;on,%although%in%2/3D%it%has%a%direc;on.%%
r
Divergence%%%%%%%%%• Divergence%is%the%change%in%the%flux%of%vector%field%QQ%such%as%gas%velocity%or%current%density.%%
• It%represents%the%source%or%destruc;on%(sink)%of%whatever%is%flowing%%
• Could%be%electric%field!%%
r·
Laplacian%• The%Laplacian%determines%the%curvature%of%a%func;on.%%
• 2nd%spa;al%deriva;ve%%
r2
CURL%• The%Curl%obtains%the%rota;on%of%a%field%(turbulence%in%water%flow)%
• Very%important%in%EM%
r⇥
Formula;on%of%Classical%Physics%• Most%of%classical%physics%(known%before%1905)%can%be%summarized%as%follows:%– Maxwell's%Equa;ons%– Conserva;on%of%charge%(can%be%deduced%from%Maxwell's%equa;ons)%
– Force%laws%– Laws%of%mo;on%– Law%of%gravita;on%
Maxwell’s%equa;ons%–%EM%fields'%Maxwell’s equations in electromagnetism are described as follows:
• The source of an electrical field is the existence of electrical charge i.e.flux of E throguh a closed surface / charge inside.
r . E = ⇢/"0 (1)
• Flux of B through a closed surface = 0, i.e. there is no magnetic monpole.
r . B = 0 (2)
• According to the Farday’s law of induction we have:
r ⇥ E = � @B
@t(3)
(A changing magnetic field will induces an electric field)
• According to Ampere’s law a current or a time varing electric field inducesa magnetic field as:
c2 r⇥B =@ E
@ t+
j
"0(4)
Maxwell’s%equa;ons%in%free%space%In free space we have ⇢ = 0 and J = 0 so we have:
r.E = 0
r . B = 0
r ⇥ E = � @B
@t
r⇥B =1
c2@ E
@ t(1)
From the math we can obtain using identities and algebra:
r ⇥ (r ⇥ A) = r(r . A) � r2 A �! (2)
r ⇥ (r ⇥ E) = r(r . E) � r2 E �!
� @
@tr ⇥ B = 0 � r2 E �!
@
@t
1
c2@E
@t= r2 E �!
r2 E � 1
c2@2E
@t2= 0 (3)
And equation 3 is the Maxwell’s equation in free space for the electric field.There is a corresponding one for the magnetic field. It is a simple wave equation.
Conserva;on%of%Charge%• Basically%conserva;on%of%charge%means%that%electrical%charge%can%not%be%created%or%destroyed.%%
• In%other%words%it%says%that%the%total%amount%of%charge%inside%any%region%can%only%change%by%the%amount%that%passes%in%or%out%of%the%region,%which%is%expressed%as%the%con;nuity%equa;on%as%follows:%
• Integral%form:%r . J +
@⇢
@t= 0
Force%Laws%• An%example%is%the%force%ac;ng%on%a%charged%par;cle%in%presence%of%electromagne;c%fields%as%given%by%Lorentz%force%equa;on:% F = q(E + v ⇥B)
Laws%of%mo;on%• According%to%classical%physics%we%have%the%force%on%moving%
par;cles%as%follows:%
F =dp
dtF = ma
m is the mass of the particle
a is the acceleration
p is the momentum
Laws%of%Gravita;on%• Newton's%law%states%that%the%force%ac;ng%on%two%par;cles%due%to%their%gravity%is%inversely%propor;onal%to%the%distance%between%them%and%is%given%by:%
%%• Where%G%is%the%gravity%constant.%
F = �Gm1 m2
r2
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