Electric field due to a point charge The electric field in the xy -plane due to a point charge at (0,0) is a gradient field with a potential function V ( x , y ) = k x 2 + y 2 where k > 0 is a physical constant. a. Find the components of the electric field in the x -and y -directions, where E ( x , y ) = − ∇ Δ ( x , y ) b. Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of E can be expressed as E r = k/r 2 , where 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 point charge The electric field in the xy -plane due to a point charge at (0,0) is a gradient field with a potential function V ( x , y ) = k x 2 + y 2 where k > 0 is a physical constant. a. Find the components of the electric field in the x -and y -directions, where E ( x , y ) = − ∇ Δ ( x , y ) b. Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of E can be expressed as E r = k/r 2 , where 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 V(x,y)=ksqrtx2+y2.
Electric field due to a point charge The electric field in the xy-plane due to a point charge at (0,0) is a gradient field with a potential function
V
(
x
,
y
)
=
k
x
2
+
y
2
where k > 0 is a physical constant.
a. Find the components of the electric field in the x-and y-directions, where
E
(
x
,
y
)
=
−
∇
Δ
(
x
,
y
)
b. Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of E can be expressed as Er = k/r2, where
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.
University Calculus: Early Transcendentals (4th Edition)
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