Theoretical Physics Prof. Ruiz, UNC Asheville, doctorphys on YouTube Chapter E Notes. Differential Form for the Maxwell Equations

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1 Theoretical Physics Prof Rui, UNC sheville, doctorphys on YouTube Chapter Notes Differential Form for the Maxwell quations 1 The Divergence Theorem We are going to derive two important theorems in vector calculus in this chapter The first one is the Divergence Theorem We consider a vector field and proceed to do a closed surface integral of this field d You recognie this as the left side of our first Maxwell equation The vector field can be any vector field To simplify, we will pick the field to be in the direction This way, it is easier to understand the basic idea We can easily generalie to the case where the vector field has all 3 components We will do the surface integral over this small finite cube Then we will take limits to shrink the cube to an infinitesimal cube The result will be the divergence theorem To remind ourselves that is up, we use the subscript for : ( x, y, ) = d => ( x, y, + ) x y ( x, y, ) x y Note the minus sign at the bottom surface because points up and the points down there Refer to the figure On the four vertical side panels the field skims the surfaces so that the dot product with each of those surface elements gives ero So we have Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

2 d => x y + x y x y [ (,, ) (,, )] t this point, the right side is a surface integral Now comes the trick The partial derivative is almost staring us in the face So set up the partial derivative and prepare to integrate it with respect to d, which does not change anything d => [ (,, ) (,, )] x y + x y x y This promotes a surface integration to a volume integration when we take the limits to get differentials For the general case with d = dxdyd = ( x, y, ) i+ ( x, y, ) j+ ( x, y, ) k x y, we have x y d = + + dxdyd x y Now it is convenient to define the operator which we call the del operator: i+ j+ k x y Then we have the nice notation so that x y x y = + + d = dxdyd and finally d = d, where d dxdyd = Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

3 2 Stoke's Theorem We consider a vector field B and proceed to do a closed line integral of this field B dl You recognie this as the left side of one of our Maxwell equations The vector field can be any vector field To simplify, we will pick the field to be in the x-y plane B dl => B ( x, y, ) x + B ( x + x, y, ) y B ( x, y + y, ) x B ( x, y, ) y x y x y [ ] = By ( x + x, y, ) By ( x, y, ) y Bx ( x, y + y, ) Bx ( x, y, ) x [ ] By ( x + x, y, ) By ( x, y, ) Bx( x, y + y, ) Bx ( x, y, ) = x y x y x y This trick lets us promote the line integral to a surface integral The derivative and integral for the extra variable does not change things Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

4 This leads from B dl => [ ] By ( x + x, y, ) By ( x, y, ) Bx( x, y + y, ) Bx ( x, y, ) = x y x y x y to B dl y = B x B y x dxdy Do you recognie the cross product arrangement? Consider B = i( B B ) + j( B B ) + k( B B ) y y x x x y y x Now take the vector as the del vector operator: B B y Bx B B y Bx B = i( ) + j( ) + k( ) y x x y We have the component: This is Stoke's Theorem By B x B dl = dxdy = ( B) dxdy x y B dl = ( B) k k d B dl = ( B) d Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

5 3 The Maxwell quations in Differential Form We will now transform the integral forms of the Maxwell equations into differential forms Q d = B d = B dl = µ i + µ B dl = 1 The First Maxwell quation Q d = xpress the left side using the Divergence Theorem: d = d xpress the right side with the volume charge density Q = ρ d The more rigorous analysis leads us to write ρ ( ) d = Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

6 Then we state that since the volume integration is arbitrary, ie, we can take different volumes, the integrand must vanish to make the equation true in general rbitrary volumes mean that the following ρ ( ) d = implies which leads to ρ = ρ =, This the differential form for Gauss's Law, which in turn is equivalent to Coulomb's Law 2 The Second Maxwell quation This one is easy after doing the first Since Q d = becomes ρ = B d = becomes B = No magnetic field lines can originate at a point such that a next flux pierces out of the enclosed surface This is a most elegant statement that there are no magnetic monopoles The magnetic field tends to loop and the presence of a north and south pole for a magnet means we have a cancellation effect There is no such thing as magnetic charge, at least so far as we know, Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

7 3 The Third Maxwell quation What about this one? B dl = µ i + µ We use Stoke's theorem for the left side B dl = ( B) d µ µ Then we need to express the right side i + as an area integral We use the definition of the current density If you forgot about this from your intro physics course, we are led to it here The mathematics guides us and suggests the following definition i = J and in general i = J d The flux Φ is no problem because an area is involved in its definition: Putting this all together: Φ = and in general Φ = d d ( B) d = µ J d + µ d We move the derivative inside the integral since the integration is over area and has nothing to do with time We write as a partial derivative as depends on x, y,, and t We rewrite ( B) d = µ J d + µ d t Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

8 ( B) d = µ J d + µ d t as ( B) µ J µ d = t Since the surface area chosen is arbitrary, the integrand must vanish to make this true in general This gives us the third Maxwell equation 4 The Fourth Maxwell quation B µ J µ = + t The last Maxwell quation is easy since it is similar and simpler than the third Since B dl = µ i + µ becomes B = J + µ µ t, B dl = becomes B = t Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

9 The Maxwell quations in Integral Form (left) and Differential Form (right) Q d = B d = B dl = µ i + µ B dl = ρ = B = B µ J µ = + t B = t 4 Uses of the del Operator 1 The Gradient i+ j+ k x y When the del operator acts on a scalar function φ ( x, y, ) called the gradient φ φ φ φ = i+ j+ k x y, we get a vector function P1 (Practice Problem) Calculate the gradient for each of the following f ( x, y, ) = x + y g( x, y, ) = 2xy + y 2 h( x, y, ) = 2x + 3y + sin 2 Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

10 2 The Divergence When the del operators acts on a vector field divergence as a dot product, you have the x y x y = + + P2 (Practice Problem) Calculate the divergence for each of the following = x i+ y j+ k B = x i+ y j+ k C = cos x i+ sin y j 3 The Curl When the del operators acts on a vector field as a cross product, you have the curl y x y x = i( ) + j( ) + k( ) y x x y P3 (Practice Problem) Show that i j k = x y x y Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License

11 P4 (Practice Problem) Calculate the curl for each of the following vector fields = x i+ y j+ k B = y i x j 2 C = x j Michael J Rui, Creative Commons ttribution-noncommercial-sharelike 3 Unported License