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