Ampere’s Law One of Maxwell’s equations for electromagnetic waves is ∇ × B = C ∂ E ∂ t , where E is the electric field, B is the magnetic field, and C is a constant. a. Show that the fields E ( z , t ) = A sin ( k z − ω t ) i B ( z , t ) = A sin ( k z − ω t ) j satisfy the equation for constants A. k, and ω , provided ω = k / C . b. Make a rough sketch showing the directions of E and B
Ampere’s Law One of Maxwell’s equations for electromagnetic waves is ∇ × B = C ∂ E ∂ t , where E is the electric field, B is the magnetic field, and C is a constant. a. Show that the fields E ( z , t ) = A sin ( k z − ω t ) i B ( z , t ) = A sin ( k z − ω t ) j satisfy the equation for constants A. k, and ω , provided ω = k / C . b. Make a rough sketch showing the directions of E and B
Solution Summary: The author explains the Maxwell's equations for magnetic waves, which satisfy the electric and magnetic fields.
Ampere’s Law One of Maxwell’s equations for electromagnetic waves is
∇
×
B
=
C
∂
E
∂
t
, where E is the electric field, B is the magnetic field, and C is a constant.
a. Show that the fields
E
(
z
,
t
)
=
A
sin
(
k
z
−
ω
t
)
i
B
(
z
,
t
)
=
A
sin
(
k
z
−
ω
t
)
j
satisfy the equation for constants A. k, and ω, provided
ω
=
k
/
C
.
b. Make a rough sketch showing the directions of E and B
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Text
Consider the vector-valued function
= (
4
cos(t) +
2
sin(t), -cos(t) +
V5
4
sin(t),
V105
f(t)%3D
20
V105
VI05 sin(e)
(a) Compute the first derivative and the nmagnitude of the first derivative.
(Hint: Make sure to simplify your answer for the magnitude as much as possible! There will be magic cancellations.)
(b) Are ^r(t) and ^r0°(t) perpendicular for all t? (Hint: You do not need to compute the second derivative. Use the
formula
(1*OF)' = (*"(t) · (1) = 26"(1) · i'(1)
and your answer from part a.)
(c) Use your answer from part b. to compute the principal normal vector in termns of the second derivative r00(E).
(Hint: Your answer should involve the symbol r 0t). Do not compute the second derivative).
University Calculus: Early Transcendentals (4th Edition)
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