MAGNETIC VECTOR POTENTIAL 5.4.1 E = 0 One of Maxwells equations, - PowerPoint PPT Presentation

MAGNETIC VECTOR POTENTIAL 5.4.1

∇× E = 0 One of Maxwell’s equations, made it useful for us E = −∇ V to define a scalar potential V, where Similarly, another one of Maxwell’s equations makes it useful for us to define the vector potential, A. Which one? A) ∇ × E = 0 ∇ ⋅ = ρ ε B ) E / 0 C ) ∇ × B = µ J 0 D ) ∇ ⋅ B = 0

� � � � ∇ × E = 0 ↔ E = − ∇ V � � ∇ × ∇ V = 0 Since � � � � � ∇ ⋅ B = ↔ B = ∇ × A 0 � � � ( ) ∇ ⋅ ∇ × A = 0 Since

MD12-3 �� The vector potential A due to a long straight wire with current I along the z-axis is in the direction parallel to: �� I � � ˆ A = ? A) z ˆ B) ϕ (azimuthal) ˆ C) s (radial) Assume Coulomb gauge

MD12-4a,b A circular wire carries current I in the xy plane. What can you say about the vector potential A at the points shown? At point a, the vector potential A is: A)Zero z B)Parallel to x-axis C)Parallel to y-axis b D)Parallel to z-axis a Assume Coulomb gauge, and A vanishes at infinity y I At point b, the vector potential A is: x A)Zero B)Parallel to x-axis C)Parallel to y-axis D)Parallel to z-axis

� � � � A ( r ) • l d What is A) The current density J B) The magnetic field B C) The magnetic flux Φ B D) It's none of the above, but is something simple and concrete E) It has no particular physical interpretation at all

The vector potential in a certain region is given by 5.19 � ˆ A(x, y) = C y x (C is a positive constant) Consider the imaginary loop shown. What can you say about the magnetic field in this region? A. B is zero B. B is non-zero, parallel to z-axis y C. B is non-zero, parallel to y-axis A D. B is non-zero, parallel to x-axis x

If the arrows represent the vector 5.24 potential A (note that |A| is the same everywhere), is there a nonzero B in the dashed region? A.Yes B.No C.Need more information to decide

BOUNDARY CONDITIONS 5.4.2

I have a boundary sheet, and would like to 6.11 learn about the change (or continuity!) of B(parallel) across the boundary. B(above) B // (above) Am I going to need to know about A) ∇× B B) ∇• B C) ???

5.28 b In general, which of the following are continuous as you move past a boundary? A) A B) Not all of A , just A perp C) Not all of A , just A // D) Nothing is guaranteed to be continuous regarding A