Field Problems: UNIZOR.COM - Physics4Teens - Waves - Field Waves
Channel: Zor Shekhtman on UNIZOR Education
Date: 2022-09-10

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Problems on Field Characteristics

Problem 1

The gravitation field is characterized by a vector of force of gravity per unit of mass of a probe object (a point-mass) as a function of its distance from the source of the field (like Earth).
The direction of this force is always towards a center of mass that is the source of gravity.

Another characteristic of a gravitational field is its potential, which is a work (a scalar) needed to bring a unit of mass from infinity to a point of interest (see "Energy" part of this "Physics 4 Teens" course, chapter "Potential Energy", topic "Gravity").

Let the source of gravitational field be at location {0,0,0} and its mass is M.
Our point of interest is at location {x,y,z}.
The gravitational potential is inversely proportional to a distance from the center of the source of gravity and can be expressed as a scalar function of three coordinates
U(x,y,z) = −G·M·(x²+y²+z²)−½
where G is a gravitational constant.

Determine the vector
grad U(x,y,z) = V(x,y,z) of gradient of this field at any point {x,y,z}, describe its direction and magnitude.

Problem 2

Given a three-dimensional vector field
V(x,y,z) = {0,0,z}

(a) Describe this vector field verbally.
(b) Is there a front surface of this field, where all vectors at it are equal?
(c) Find a divergence of this field.
(d) Is there a part of space where divergence is zero?
(e) Find a curl of this field.

Problem 3

Given a two-dimensional vector field
V(x,y) = {−y/√(x²+y²), x/√(x²+y²)}

(a) Describe this vector field verbally.
(b) Is there a front line of this field, where all vectors at it are equal?
(c) Find a divergence of this field.
(d) Is there a part of space where divergence is zero?
(e) Find a curl of this field.

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Solutions are in the video-lecture and on
UNIZOR.COM - Physics4Teens - Waves - Field Waves - Field Problems