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
Integrals of Line and Work
A cyclist rides up a mountain along the path shown in the figure. She makes one complete revolution around the mountain in reaching the top, while her vertical rate of climb is constant. Throughout the trip, she exerts a force described by the vector field
F(x,y,z) = z2i + 3y2j + 2xk
What is the work done by the cyclist in travelling from A to B?
Calculus Answer, please don't use italics, as I do not understand it
Calculate the work that a constant force field F does on a particle that moves uniformly once along the path of the curve x2 + y2 = 1.How much is the work now worth if we take F(x, y) = (αx, αy), where α is any positive constant?
Work in force fields Find the work required to move an object in the following force field along a line segment between the given points. Check to see whether the force is conservative.
F = ex + y ⟨1, 1, z⟩ from A(0, 0, 0) to B(-1, 2, -4)
University Calculus: Early Transcendentals (4th Edition)
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