lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the Maxwell equations into diffeential fom Q da B da B dl µ + µ B dl 1 The Fist Maxwell quation Q da xpess the left side using the Divegence Theoem da dv V xpess the ight side with the volume chage densit Q V ρ dv Then, V dv V ρ dv A igoous analsis equies us to wite it this wa: V ρ ( ) dv Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
Then we state that since the volume integation is abita, ie, we can take diffeent volumes, the integand must vanish to make the equation tue in geneal Abita volumes mean that the following V ρ ( ) dv implies which leads to ρ ρ, This the diffeential fom fo Gauss's Law, which in tun is equivalent to Coulomb's Law The Second Maxwell quation This one is eas afte doing the fist Since Q da becomes ρ B da becomes B No magnetic field lines can oiginate at a point since thee ae no magnetic monopoles Theefoe, thee ae no diveging magnetic field lines fom a point This is a most elegant statement that thee ae no magnetic monopoles The magnetic field tends to loop and the pesence of a noth and south pole fo a magnet means we have a cancellation effect n othe wods, thee is no such thing as magnetic chage, at least so fa as we know f we eve find a magnetic monopole, then this basic equation will have to be modified And if so, which othe Maxwell equation needs to be modified to account fo a cuent of moving monopoles? You answe can be checked b pefect smmet in the Maxwell equations: chage, electical cuent, monopoles (magnetic chage), and magnetic cuents Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License,
3 The Thid Maxwell quation What about this one? B dl µ + µ We use Stoke's theoem fo the left side B dl ( B) da A µ µ Then we need to expess the ight side + as an aea integal We use the definition of the cuent densit f ou ae haz on this fom ou into phsics couse, we ae led to it hee The mathematics guides us and suggests the following definition: JA and in geneal J da A The flux Φ is no poblem because an aea is involved in its definition alead: Putting this all togethe: Φ A and in geneal Φ da d ( B) da µ J da + µ da A A A We move the deivative inside the integal since the integation is ove aea and has nothing to do with time We wite as a patial deivative as depends on x,, z, and t ( B) da µ J da + µ da A A A A Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
We now ewite ( B) da µ J da + µ da A A A as A ( B) µ J µ da Since the suface aea chosen is abita, the integand must vanish to make this tue in geneal This gives us the thid Maxwell equation 4 The Fouth Maxwell quation B µ J µ + The last Maxwell quation is eas since it is simila and simple than the thid Since B dl µ + µ becomes B J + µ µ, B dl becomes B t Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
The Maxwell quations in ntegal Fom (left) and Diffeential Fom (ight) Q da B da B dl µ + µ B dl ρ B B µ J µ + B t nsight into the Divegence Let's see if we can gain some insight into the divegence b investigating whee we have a point chage Theefoe, 1 1 4π ρ We want to do the calculation in Catesian coodinates, so we expess in tems of i,, and k Coutes Andeggs, Wikimedia Fom the left figue ou see we can fom the vecto along b setting x i+ + z k The unit vecto along the adial diection is then x z i+ + k Note that 1 Wh? Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
1 1 4π The electic field with x z i+ + k becomes 1 1 x z 1 x z i k i k 4π + + + + 4π 3 3 3 x + + z now, but it is best not to do this in ode to keep ou We could use notation concise Now we ae ead to take the divegence x z 1 x z + + ( ) + ( ) + ( ) 3 3 3 x z 4π x z The thee deivatives ae simila so wok with the fist one x 1 1 1 1 ( ) + x + x ( ) 3 3 3 3 3 x x x 1 3x x ( ) The two pats of the second tem ae 3 4 and 1 1 x x + + z ( x) x x x + + z P1 (Pactice Poblem) Show this quickl b implicit diffeentiation of Putting it all togethe, x 1 3x ( ) 3 3 5 x Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License Finall we get the divegence below 1 1 3x 3 3z 1 3 3 3 3 5 5 5 3 3 4π 4π ρ But Wait! Wh didn't we get as thee is chage somewhee? You will see wh Read on
Let's t inside a unifom sphee of chage Fom befoe we know Q d A, 1 4 (4 π ) ρ π 3, and 3 ρ 3 Then, ρ x z ρ i k x i z k 3 + + + + 3 x z ρ x z ρ ρ + + + + (3) x z 3 x z 3 Now we get ρ and that's because chage densit is actuall at ou location This is a pofound point! We now coectl undestand the fist Maxwell equation! Outside the chage in space, awa fom the chage, ou get even though ou have an electic field out thee This is a deep discove into the meaning of the diffeential fom fo Gauss's Law n empt space, ou get zeo, but when ou ae in the chage-densit egion ou get the nonzeo value Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
3 nsight into the Cul Let's see if we can gain some insight into the cul b investigating B µ J µ + whee We onl have cuent in a wie mage Cedit: Wikimedia, fom Use: Stanneed fom an oiginal b Use: Wapcaplet Recall ou magnetic field poduced b a cuent in a wie B µ θ B µ J We will calculate the cul in calculation in Catesian coodinates This means we need to expess θ in tems of i and You find the usual Catesian unit vectos i and in the left figue as well as the pola unit vectos and θ, All these unit vectos point in inceasing diections of thei espective coodinates Fom the ight tiangle in ed, we aive at the expession of θ in tems of i and : θ sinθ i+ cosθ, which also has the equied unit length Wh? Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
While we ae hee, we can obtain the esult cosθ i+ sinθ Ou two equations ae below cosθ i+ sinθ θ sinθ i+ cosθ θ θ and θ Note that 1 Below ae ou egula tansfomations between pola and Catesian coodinates which ou encounteed in math at some point befoe x cosθ sinθ x + θ tan 1 x P (Pactice Poblem) Find i and in tems of and θ xpessing B µ θ with θ sinθ i+ cosθ, we obtain µ B sinθ i+ cosθ Now use x cosθ and sinθ, ie, cos x θ and sin θ µ x B i+ µ x B i+ Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
Take the cul Note that in this case is the pola coodinate and not the spheical coodinate we encounteed in the Gaussian analsis i k ( ) ( ) x z x x µ µ x B k + P3 (Pactice Poblem) Wh is thee onl a z-component fo this cul? x 1 1 1 ( ) + x ( ) + x( ) x x x Fist conside 3 The last deivative is x x, using x + and d xdx + d This last step is the implicit-diffeentiation tick in two dimensions x and Then x 1 x 1 x ( ) + x( )( ) x 3 4 and µ 1 x 1 + π B k 4 4 µ ( x + ) µ B k k 4 4 B µ J But Wait! Wh didn't we get You will see wh Read on as thee is cuent somewhee? Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License
Let's t inside a unifom wie of cuent Fom befoe we know B dl µ, B( π ) µ Jπ Using θ sinθ i+ cosθ, we have, and B µ J θ µ J µ J B θ sinθ i+ cosθ µ J µ J B sinθ i+ cosθ i+ x Then, i k µ J µ J x µ J B k + k x z x x B µ J k B µ J We now get the nonzeo cuent densit since we ae at a point whee cuent densit actuall exists We now coectl undestand the Maxwell equation wit cuent souces! Michael J Ruiz, Ceative Commons Attibution-NonCommecial-ShaeAlike 3 Unpoted License