For the following exercises, assume that an electric field in the xy -plane caused by an infinite line of charge along the x -axis is a gradient field with potential function V(x, y ) = c In ( r 0 x 2 + y 2 ) , where c > 0 is a constant and r 0 is a reference distance at which the potential is assumed to be zero. 29. Show that the electric field at a point in the xy -plane is directed outward from the origin and has magnitude El = c r ,. where r = x 2 = y 2 A flow line (or sweamline) of a vector field F is a curve r( t ) such that dr/dt = F(r( t )). If F represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve c( t ) is a flow line of the given velocity vector field F ( x , y,z ).
For the following exercises, assume that an electric field in the xy -plane caused by an infinite line of charge along the x -axis is a gradient field with potential function V(x, y ) = c In ( r 0 x 2 + y 2 ) , where c > 0 is a constant and r 0 is a reference distance at which the potential is assumed to be zero. 29. Show that the electric field at a point in the xy -plane is directed outward from the origin and has magnitude El = c r ,. where r = x 2 = y 2 A flow line (or sweamline) of a vector field F is a curve r( t ) such that dr/dt = F(r( t )). If F represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve c( t ) is a flow line of the given velocity vector field F ( x , y,z ).
For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function V(x, y) = c In
(
r
0
x
2
+
y
2
)
,
where c > 0 is a constant and r0is a reference distance at which the potential is assumed to be zero.
29. Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude El =
c
r
,. where r =
x
2
=
y
2
A flow line (or sweamline) of a vector field F is a curve r(t) such that dr/dt = F(r(t)). If F represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve c(t) is a flow line of the given velocity vector field F(x, y,z).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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