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.
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.a. Find the gradient field F = ∇φ.b. Evaluate F at the points A, B, C, and D.c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.
φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)
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
Example: Sketch the direction field for the equation y' = y – t over the
square -2 < t, y < 2, then using this direction field sketch the solution that
passes through the points (-1,±1).
University Calculus: Early Transcendentals (4th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY