Any charge that is in an electric field is affected by force. In this regard, when the charge moves in the field, a certain work of the electric field occurs. How to calculate this work?
The work of the electric field consists in the transfer of electric charges along the conductor. It will be equal to the product of voltage, current and time spent on work.
Applying the formula of Ohm's law, we can get several different versions of the formula for calculating the current work:
A = U˖I˖t = I²R˖t = (U² / R) ˖t.
In accordance with the law of conservation of energy, the work of the electric field is equal to the change in the energy of a single section of the circuit, and therefore the energy released by the conductor will be equal to the work of the current.
Express in the SI system:
[A] = B˖A˖s = W˖s = J
1 kWh = 3,600,000 J.
Let's make an experiment. Consider the movement of a charge in a field of the same name, which is formed by two parallel plates A and B and charged unlike charges. In such a field, the lines of force are perpendicular to these plates along their entire length, and when plate A is positively charged, then the field strength E will be directed from A to B.
Suppose that the positive charge q has moved from point a to point b along an arbitrary path ab = s.
Since the force that acts on the charge that is in the field will be F = qE, the work done when the charge moves in the field according to the given path will be determined by the equality:
A = Fs cos α, or A = qFs cos α.
But s cos α = d, where d is the distance between the plates.
It follows: A = qEd.
Suppose now the charge q moves from a and b in fact acb. The work of the electric field, accomplished along this path, is equal to the sum of the works performed in individual sections of it: ac = s₁, cb = s₂, i.e.
A = qEs₁ cos α₁ + qEs₂ cos α₂,
A = qE (s₁ cos α₁ + s₂ cos α₂,).
But s₁ cos α₁ + s₂ cos α₂ = d, and therefore, in this case, A = qEd.
In addition, suppose that the charge q moves from a to b along an arbitrary curve of the line. In order to calculate the work done on this curved path, it is necessary to stratify the field between plates A and B with a number of parallel planes that will be so close to each other that individual sections of the path s between these planes can be considered straight.
In this case, the electric field work performed on each of these segments of the path will be equal to A₁ = qEd₁, where d₁ is the distance between two adjacent planes. And the full work along the entire path d will be equal to the product of qE and the sum of the distances d₁ equal to d. Thus, as a result of the curved path, the perfect work will be equal to A = qEd.
The examples considered by us show that the work of the electric field to move the charge from any point to another does not depend on the shape of the path of movement, but depends solely on the position of these points in the field.
In addition, we know that the work that is done by gravity when moving the body along an inclined plane having a length l will be equal to the work that the body does when falling from a height h and the height of the inclined plane. This means that the work of gravity, or, in particular, the work when moving the body in the field of gravity, also does not depend on the shape of the path, but depends only on the difference in height of the first and last points of the path.
So it can be proved that such an important property can possess not only a uniform, but also any electric field. Gravity also has a similar property.
The work of the electrostatic field in moving a point charge from one point to another is determined by the linear integral:
A₁₂ = ∫ L₁₂q (Edl),
where L₁₂ is the trajectory of the charge, dl is the infinitely small displacement along the trajectory. If the contour is closed, then the symbol ∫ is used for the integral; in this case, it is assumed that the bypass path is selected.
The work of electrostatic forces does not depend on the shape of the path, but only on the coordinates of the first and last points of displacement. Consequently, the strength of the field is conservative, and the field itself is potentially. It is worth noting that the work of any conservative force in a closed path will be zero.