Elec%4705%Lec%2% Classical%Physics%

Transcription

1 Elec%4705%Lec%2% Classical%Physics%

2 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.%

3 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% %

4 The%``Del''%operator%QQ%%% Vector%operator!% A%summary%of%some%mathema;cal% opera;ons% operation name result r r r V Gradient Vector r. V Divergence Scaler r 2 V Laplacian Scaler r V Curl Vector

5 r 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.%%

6 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!%%

7 r 2 Laplacian% The%Laplacian%determines%the%curvature% of%a%func;on.%% 2 nd %spa;al%deriva;ve%%

8 r CURL% The%Curl%obtains%the%rota;on%of%a%field% (turbulence%in%water%flow)% Very%important%in%EM%

9 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%

10 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. According to the Farday s law of induction we have: r.b = 0 (2) r (A changing magnetic field will induces an electric field) (3) According to Ampere s law a current or a time varing electric field induces a magnetic field as: c 2 r B E + j t " 0

11 Maxwell s%equa;ons%in%free%space% In free space we have = 0 and J =0sowehave: r.e =0 r.b =0 r r B = E c t From the math we can obtain using identities and algebra: (1) r (r A) =r(r.a) r 2 A! (2) r (r E) =r(r.e) r r B =0 c r 2 E = r 2 E! 2 E c 2 =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.

12 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:% =0

13 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)

14 Laws%of%mo;on% According%to%classical%physics%we%have%the%force%on%moving% par;cles%as%follows:% F = dp dt F = ma m a p is the mass of the particle is the acceleration is the momentum

15 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:% F = G m 1 m 2 r 2 % % Where%G%is%the%gravity%constant.%