Download UIUC Physics 435 Lecture Notes 3: Electric and Magnetic Fields and Sources I and more Study notes Guiding Electromagnetic Systems in PDF only on Docsity!
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1
Questions for P435 Lecture Notes 3
1.) What two mathematical conditions uniquely specifies the nature of an arbitrary, but
differentiable vector field F ( r )
G G
that goes to zero faster than 1/ r as r → ∞.
Answer: ∇ F ( r )
G G G
i and ∇ × F ( r )
G G G
. Why??? See/read Griffiths Appendix B r.e. the Helmholtz theorem, and in particular, the corollary to the Helmholtz theorem, p. 557.
2.) Is the absolute potential V ( r^ G^ )at an arbitrary point in space ( r^ G^ )a physically meaningful
quantity? Or are only potential differences Δ V ab ≡ V ( rb ) − V ( ra )physically meaningful?
3.) Know/understand the concept of equipotential surfaces. Especially note/understand that
contours of constant potential are perpendicular to electric field lines E r ( )
G G
everywhere in space!
4.) Know/understand the relation between electric field E r ( )
G G
and the scalar potential V ( r^ G^ ),
namely E r ( ) = −∇ V ( r )
G G G
(in differential form)
and ( ) ( ) ( )
b
Δ V ab ≡ V rb − V ra = − ∫ a E r d
G G G
i A (in integral form).
5.) Know/understand how Poisson’s equation ∇ 2 V ( r^ G^ )= − ρ encl ε o and Laplace’s equation
∇ 2 V^ ( r^ G^ )= 0 are obtained by using the relations ∇ E r ( )
G G G
i , ∇ × E r ( )
G G G
and E r ( ) = −∇ V ( r )
G G G
6.) How is the boundary condition Eabove ⊥^ − Ebelow ⊥ = σ ε 0 obtained?
What is the physical meaning of this boundary condition?
7.) How is the boundary condition Eabove &^ − Ebelow &^ = 0 obtained? What is the physical meaning of this boundary condition?
8.) How is the boundary condition 0
above below interface interface
V V
n n
σ ε
∂ ∂^ ⎛^ ⎞
obtained?
What is the physical meaning of this boundary condition?
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2
Answers to Questions for P435 Lecture Notes 3
1.) What two mathematical conditions uniquely specifies the nature of an arbitrary, but
differentiable vector field F ( r )
G G
that goes to zero faster than 1/ r as r → ∞.
Answer: ∇ F ( r )
G G G
i and ∇ × F ( r )
G G G
fully specify the physics nature of the vector field F ( r )
G G
for the class of differentiable vector functions F ( r )
G G
that go to zero faster than 1/ r as r → ∞. Please see/read Griffiths Appendix B r.e. the Helmholtz theorem, and in particular, the corollary to the Helmholtz theorem, p. 557.
2.) Is the absolute potential V ( r^ G^ )at an arbitrary point in space ( r^ G^ )a physically meaningful
quantity? Or are only potential differences Δ V ab ≡ V ( rb ) − V ( ra )physically meaningful?
Indeed, only differences in potential Δ V ab ≡ V ( rb ) − V ( ra )are physically meaningful; the
absolute potential V ( r^ G^ )at an arbitrary point in space ( r^ G^ )has no physical meaning.
This aspect of the (scalar) potential V ( r^ G^ )actually goes quite deep, into the heart of
electromagnetism, for it is intimately connected to the so-called gauge invariant nature of the electromagnetic interaction. One can always add an arbitrary constant to the potential
V ( r )
G (^) and this will/can have no physically observable consequences. We will also see/learn
later on in the P436 course that one can also add an arbitrary gradient of a scalar function
∇ λ ( r )
G G
to the (so-called magnetic) vector potential A r ( )
G G
also with no physically observable
consequences – please see/read P436 Lecture Notes 16. The scalar potential V ( r^ G^ )and the
vector potential A r ( )
G G
respectively are the temporal and spatial components of the
relativistic four-vector potential A^ μ^ ( r ) ≡ ( V ( r ) c A r , ( )) = ( A^0^ , A^1^ , A^2^ , A^3^ ) =( A^0 , Ax , Ay , Az )
G G G G
3.) Know/understand the concept of equipotential surfaces. Especially note/understand that
contours of constant potential are perpendicular to electric field lines E r ( )
G G
everywhere in space!
Equipotential surfaces are everywhere perpendicular to electric field lines E r ( )
G G
because of /
due to the fact that E r ( ) = −∇ V ( r )
G G G G
4.) Know/understand the relation between electric field E r ( )
G G
and the scalar potential V ( r^ G^ ),
namely E r ( ) = −∇ V ( r )
G G G
(in differential form)
and ( ) ( ) ( )
b
Δ V ab ≡ V rb − V ra = − ∫ a E r d
G G G
i A (in integral form).
Please see/read P435 Lecture Notes 3, p. 2-3.
