Electric field due to a line of charge The electric field in the xy -plane due to an infinite line of charge along the z -axis is a gradient field with a potential function V ( x , y ) = c ln ( 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 0 (see figure). a. Find the components of the electric field in the x- and y directions, where E ( x , y ) = − ∇ V ( x , y ) . b. Show that the electric field at a point in the xy- plane is directed outward from the origin and has magnitude | E | = c / r , . where r = x 2 + y 2 . c. Show that the vector field is orthogonal to the equipotential curves at all points in the domain of V.
Electric field due to a line of charge The electric field in the xy -plane due to an infinite line of charge along the z -axis is a gradient field with a potential function V ( x , y ) = c ln ( 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 0 (see figure). a. Find the components of the electric field in the x- and y directions, where E ( x , y ) = − ∇ V ( x , y ) . b. Show that the electric field at a point in the xy- plane is directed outward from the origin and has magnitude | E | = c / r , . where r = x 2 + y 2 . c. Show that the vector field is orthogonal to the equipotential curves at all points in the domain of V.
Solution Summary: The author calculates the gradient field of the potential function E(x,y)=cmathrmlnleft.
Electric field due to a line of charge The electric field in the xy-plane due to an infinite line of charge along the z-axis is a gradient field with a potential function
V
(
x
,
y
)
=
c
ln
(
r
0
x
2
+
y
2
)
.
where c > 0 is a constant and r0 is a reference distance at which the potential is assumed to be 0 (see figure).
a. Find the components of the electric field in the x-and y directions, where
E
(
x
,
y
)
=
−
∇
V
(
x
,
y
)
.
b. Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude
|
E
|
=
c
/
r
,
.where
r
=
x
2
+
y
2
.
c. Show that the vector field is orthogonal to the equipotential curves at all points in the domain of V.
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.
Determine whether the functions y, and y, are linearly dependent on the interval (0,1)
y=1-2 sint, y, = 12 cos 2t
Select the correct choice below and, if necessary, fill in the answer box within your choice.
O A. Since y, = ( )y½ on (0,1), the functions are linearly dependent on (0, 1).
(Simplify your answer.)
O B. Since y,= ( )y½ on (0,1), the functions are linearly independent on (0,1).
(Simplify your answer ).
OC. Since y, is not a constant multiple of y, on (0,1), the functions are linearly independent on (0, 1)
O D. Since y, is not a constant multiple of y, on (0,1), the functions are linearly dependent on (0,1).
Determine whether the functions y,
and y, are linearly dependent on the interval (0,1).
y1 =sint cos t, y, = 5 sin 2t
Select the correct choice below and, if necessary, fill in the answer box within your choice.
O A. Since y, = y, on (0,1), the functions are linearly dependent on (0,1).
(Simplify your answer.)
O B. Since y, =O2 on (0,1), the functions are linearly independent on (0,1)
(Simplify your answer.)
O C. Since y, is not a constant multiple of y, on (0,1), the functions are linearly independent on (0,1).
O D. Since y, is not a constant multiple of y, on (0,1), the functions are linearly dependent on (0,1).
Determine whether the functions y, and y, are linearly dependent on the interval (0,1).
V=1-2 sin t, y2 = 12 cos 2t
Select the correct choice below and, if necessary, fill in the answer boX within your choice.
CA Since y,= ]y on (0,1), the functions are linearly dependent on (0,1).
(Simplify your answer ).
OB. Since y = ( )y, on (0,1), the functions are linearly independent on (0,1).
(Simplify your answer)
OC. Since y, is not a constant multiple of y, on (0 1). the functions are linearly independent on (0,1)
OD. Since y, is not a constant multiple of y, on (0,1), the functions are linearly dependent on (0, 1).
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