Field, Potential
The prototype for an abstract concept of a field presented below is a gravitational field. The force of this field acting on a unit mass is a prototype for a concept of field intensity.
On UNIZOR.COM these concepts were introduced in the Physics 4 Teens course, part Energy, chapters Energy of Gravitational Field and Gravitational Potential.
Here we will formally define these and other concepts and derive a few important field properties.
Field is an area (a subset) of points P{x,y,z} in our three-dimensional space with a vector of force F(P) called field intensity defined at each point P of this area and a real function U(P) called potential defined at exactly the same points, when the following equation between a force and a potential at each point is held:
∀P{x,y,z}: F(P) = −∇U(P)
where symbol ∇ signifies gradient of a function U - a vector of function's partial derivatives by each coordinate ∇U(P) = ∇U(x,y,z) =
= ||∂U/∂x,∂U/∂y,∂U/∂z||
It should be noted that in many cases authors do not differentiate between the field and the field intensity force, defining the field as the force, which in our opinion is misleading.
That's why we define field as an area of a space, which corresponds to the usual meaning of this word, and field intensity as a force acting inside this area.
Let's assume that a material point acted upon by the force of a field is moving during the time t period from t=t1 to t=t2 from point A to point B along some trajectory
P(t) = {x(t),y(t),z(t)}
where P(t1)=A and P(t2)=B
The first important property of a field is that the work of the field intensity force F(P) along any trajectory of an object moving in the field depends only on the field potential at the beginning and at the end of a trajectory and is independent of a path between these two points. In other words, no matter how an object that experiences a field force moves from point A to point B, the work of the force remains the same and depends only on the field potential at end points A and B.
Here is the proof of this statement.
Assume, at moment t1 our object is at point A, and it moves along a trajectory
The field intensity F is defined for all points of a field, including the points along the trajectory of an object and is equal to
Work performed by any force during the observed time is, by definition,
Wt∈[t1,t2] = ∫t∈[t1,t2]F(t)·dr(t)
In our case of a field force
F(t) = F(x(t),y(t),z(t)) =
= −∇U(x(t),y(t),z(t)) =
= −||∂U/∂x,∂U/∂y,∂U/∂z||
and
dr(t)=||dx(t),dy(t),dz(t)||
Let's evaluate the scalar product of two vectors under the integral
||∂U/∂x,∂U/∂y,∂U/∂z||·
||dx(t),dy(t),dz(t)|| =
= (∂U/∂x)·dx(t) +
+ (∂U/∂y)·dy(t) +
+ (∂U/∂z)·dz(t) =
= dU(x(t),y(t),z(t)) = dU(t)
Above is a full differential (infinitesimal increment) of function U(t)=U(x(t),y(t),z(t)) on infinitesimal time interval from t to t+dt.
Therefore, the work performed by a field force F(t) equals to
Wt∈[t1,t2] = −∫t∈[t1,t2]dU(t) =
= −U(t2) + U(t1) = U(t1) − U(t2)
As we see, the total amount of work performed by a field intensity force depends only on the field potentials at end points and does not depend on the path (trajectory) an object took to reach from start to finish.
An obvious consequence of this property of the field is that an amount of work the field intensity force performed when an object moves along a trajectory with the finishing point coinciding with the starting one equals to zero.
Recall that amount of work the field intensity force performed when an object moves from one point to another equals to an increment of the object's kinetic energy (see the previous lecture Newton's Laws of this chapter of a course)
Wt∈[t1,t2] = T(t2) − T(t1)
Therefore,
T(t2) − T(t1) = U(t1) − U(t2)
from which follows the Law of Conservation of Energy
T(t1) + U(t1) = T(t2) + U(t2)
It states that the sum of kinetic and potential energy is not changing during an object's movement within a field.
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