Dr. Hugh Blanton ENTC 3331

Gauss’s Law

Dr. Blanton - ENTC Gauss’s Theorem 3 Recall Divergence literally means to get farther apart from a line of path, or To turn or branch away from.

Dr. Blanton - ENTC Gauss’s Theorem 4 Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles: Goes straight ahead at constant velocity.  (degree of) divergence  0

Dr. Blanton - ENTC Gauss’s Theorem 5 Now suppose they turn with a constant velocity  diverges from original direction (degree of) divergence  0

Dr. Blanton - ENTC Gauss’s Theorem 6 Now suppose they turn and speed up.  diverges from original direction (degree of) divergence >> 0

Dr. Blanton - ENTC Gauss’s Theorem 7 Current of water  No divergence from original direction (degree of) divergence = 0

Dr. Blanton - ENTC Gauss’s Theorem 8 Current of water  Divergence from original direction (degree of) divergence ≠ 0

Dr. Blanton - ENTC Gauss’s Theorem 9 Source Place where something originates. Divergence > 0.

Dr. Blanton - ENTC Gauss’s Theorem 10 Sink Place where something disappears. Divergence < 0.

Dr. Blanton - ENTC Gauss’s Theorem 11 Derivation of Divergence Theorem Suppose we have a cube that is infinitesimally small. one of six faces Vector field, V(x,y,z) x y z

Dr. Blanton - ENTC Gauss’s Theorem 12 Need the concept of flux: water through an area current through an area water flux per cross-sectional area (flux density implies (total) flux = = scaler. A

Dr. Blanton - ENTC Gauss’s Theorem 13 Let’s assume the vector, V(x,y,z), represents something that flows, then flux through one face of the cube is: For example might be: and

Dr. Blanton - ENTC Gauss’s Theorem 14 The following six contributions for each side of the cube are obtained:

Dr. Blanton - ENTC Gauss’s Theorem 15 Now consider the opposite faces of the infinitesimally small cube. This holds equivalently for the two other pairs of faces. x y z differential change of V x over dx vector magnitude on the input side.

Dr. Blanton - ENTC Gauss’s Theorem 16 x y z and Flux in the x-direction.

Dr. Blanton - ENTC Gauss’s Theorem 17 Divergence Theorem Gauss’s Theorem Valid for any vector field Valid for any volume, Whatever the shape. Note that the above only applies to the Cartesian coordinate system.

Dr. Blanton - ENTC Gauss’s Theorem 18 Since Gauss’s law can be applied to any vector field, it certainly holds for the electric field, and the electric flux density,. The use of in this context instead of is historical.

Dr. Blanton - ENTC Gauss’s Theorem 19 If Gauss’s law is true in general, it should be applicable to a point charge. Constuct a virtual sphere around a positive charge with radius, R. must be radially outward along the unit vector,. + q

Dr. Blanton - ENTC Gauss’s Theorem 20

Dr. Blanton - ENTC Gauss’s Theorem 21 What about the volume integral? only has a component along the radius vector

Dr. Blanton - ENTC Gauss’s Theorem 22 What is this?

Dr. Blanton - ENTC Gauss’s Theorem 23 Throw in some physics! integration and differentiation cancel out

Dr. Blanton - ENTC Gauss’s Theorem 24 So what? Coulomb’s law and Gauss’s law are equivalent for a point charge!

Dr. Blanton - ENTC Gauss’s Theorem 25 divergence theorem

Dr. Blanton - ENTC Gauss’s Theorem 26 Gauss’s Law

Dr. Blanton - ENTC Gauss’s Theorem 27 Because of its greater mathematical versatility, Gauss’s law rather than Coulomb’s law is a fundamental postulate of electrostatics. A postulate is believed to be true, although no proof may be possible. Any surface of an arbitrary volume.

Dr. Blanton - ENTC Gauss’s Theorem 28 Note which infers definition of charge distribution Gauss’s Law Differential form of Gauss’s Law

Dr. Blanton - ENTC Gauss’s Theorem 29 Maxwell Equation One of two Maxwell equations for electrostatics.

Dr. Blanton - ENTC Gauss’s Theorem 30

Dr. Blanton - ENTC Gauss’s Theorem 31 Electric flux density or Displacement Field [C/m 2 ] Charge Density [C] Magnetic Induction [Weber/m 2 or Tesla]] Magnetic Field [A/m] Current Density [A/m 2 ] Electric Field [V/m] Time [s] Page 139

Dr. Blanton - ENTC Gauss’s Theorem 32 Page 139

Dr. Blanton - ENTC Gauss’s Theorem 33 Use Gauss’s law to obtain an expression for the E-field from an infinitely long line of charge. 0 X

Dr. Blanton - ENTC Gauss’s Theorem 34 Symmetry Conditions Infinite line of charge

Dr. Blanton - ENTC Gauss’s Theorem 35 Gauss’s law considers a hypothetical closed surface enclosing the charge distribution. This Gaussian surface can have any shape, but the shape that minimizes our calculations is the shape often used. 0

Dr. Blanton - ENTC Gauss’s Theorem 36 The total charge inside the Gaussian volume is: The integral is: The right and left surfaces do not contribute since.

Dr. Blanton - ENTC Gauss’s Theorem 37 and

Dr. Blanton - ENTC Gauss’s Theorem 38 Two infinite lines of charge. Each carrying a charge density,  l. Each parallel to the z-axis at x = 1 and x = -1. What is the E-field at any point along the y-axis?

Dr. Blanton - ENTC Gauss’s Theorem 39 For a single line of constant charge Using the principle of superposition of fields:

Dr. Blanton - ENTC Gauss’s Theorem 40 x x y z 1

Dr. Blanton - ENTC Gauss’s Theorem 41 Only interested in the y-component of the field

Dr. Blanton - ENTC Gauss’s Theorem 42 A spherical volume of radius a contains a uniform charge density  V. Determine for and + q Note: Charge distribution for an atomic nucleus where a = 1.2  m  A ⅓ (A is the mass number)

Dr. Blanton - ENTC Gauss’s Theorem 43 Outside the sphere (R  a), use Gauss’s Law To take advantage of symmetry, use the spherical coordinates: and

Dr. Blanton - ENTC Gauss’s Theorem 44 Field is always perpendicular for any sphere around the volume. The left hand side of Gauss’s Law is

Dr. Blanton - ENTC Gauss’s Theorem 45 Recall that

Dr. Blanton - ENTC Gauss’s Theorem 46

Dr. Blanton - ENTC Gauss’s Theorem 47 Inside the sphere (R  a), use Gauss’s Law previously calculated

Dr. Blanton - ENTC Gauss’s Theorem 48

Dr. Blanton - ENTC Gauss’s Theorem 49 Thin spherical shell Find E-field for and

Dr. Blanton - ENTC Gauss’s Theorem 50 Inside ( ) Gauss’s Law This is only possible if.

Dr. Blanton - ENTC Gauss’s Theorem 51 Outside ( ) Gauss’s Law previously calculated

Dr. Blanton - ENTC Gauss’s Theorem 52

Dr. Blanton - ENTC Gauss’s Theorem 53 An electric field is given as Determine Q in a 2m  2m  2m cube.

Dr. Blanton - ENTC Gauss’s Theorem 54 Maxwell’s equation of Electrostatics z x y

Dr. Blanton - ENTC Gauss’s Theorem 55 z x y For the surface 1 directed in the x-direction. 1

Dr. Blanton - ENTC Gauss’s Theorem 56 z x y 1 2 For the surface 2 directed in the -x-direction.

Dr. Blanton - ENTC Gauss’s Theorem 57 z x y 3 For the surface 3 & 4 directed in the z- & -z directions. 4

Dr. Blanton - ENTC Gauss’s Theorem 58 z x y For the surface 5 directed in the y-direction. 5

Dr. Blanton - ENTC Gauss’s Theorem 59 z x y For the surface 6 directed in the -y-direction. 6

Dr. Blanton - ENTC Gauss’s Theorem 60 By superposition Indeed, there is no charge in the cube.

Dr. Blanton - ENTC Gauss’s Theorem 61 Find in all regions of an infinitely long cylindrical shell. Inner shell( ) Cylindrical volume. 3 1

Dr. Blanton - ENTC Gauss’s Theorem 62 Shell itself ( ) Cylindrical coordinates. 3 1

Dr. Blanton - ENTC Gauss’s Theorem 63 Top and bottom face of cylinder do not contribute to.

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Dr. Blanton - ENTC Gauss’s Theorem 65