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.
Suppose f(x,y)=x/y, P=(0,−1) and v=3i+3j.
A. Find the gradient of f.∇f= ____i+____jNote: Your answers should be expressions of x and y; e.g. "3x - 4y"
B. Find the gradient of f at the point P.(∇f)(P)= ____i+____j Note: Your answers should be numbers
C. Find the directional derivative of f at P in the direction of v.Duf=?Note: Your answer should be a number
D. Find the maximum rate of change of f at P.maximum rate of change of f at P=?
Note: Your answer should be a number
E. Find the (unit) direction vector in which the maximum rate of change occurs at P.u= ____i+____jNote: Your answers should be numbers
Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge −1 at a position x between them. It follows from Coulomb's Law that the net force acting on the middle particle is
Suppose f(x,y)=x/y, P=(1,−1) and v=−4i−4j.
A. Find the gradient of f. ∇f= ____i+____jNote: Your answers should be expressions of x and y; e.g. "3x - 4y"
B. Find the gradient of f at the point P. (∇f)(P)= ____i+____j Note: Your answers should be numbers
C. Find the directional derivative of f at P in the direction of v. Duf=?Note: Your answer should be a number
D. Find the maximum rate of change of f at P. maximum rate of change of f at P=?Note: Your answer should be a number
E. Find the (unit) direction vector in which the maximum rate of change occurs at P. ?= ____i+____jNote: Your answers should be numbers
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
University Calculus: Early Transcendentals (4th Edition)
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