§ 7 Work of the electrostatic field intensity at moving charge.

Potential nature of the field forces.

Circulation of intensity

Consider the electrostatic field produced by the charge q. Let it move a test charge q0. At any point of the field on the charge q0 a force


where  -force module ,  - the unit vector of the radius vector , which determines the position of the charge q0 relative to the charge q. Since the force varies from point to point, the work force of the electrostatic field can be written as a variable work force:





Because the regarded charge transport from point 1 to point 2 along an arbitrary trajectory, it can be concluded that the work on the movement of a point charge in an electric field does not depend on the shape of the path but is determined only the initial and final positions of the charge. This indicates that the electrostatic field is potential, and the strength of Coulomb - conservative force. Work on moving charge in a field along a closed path is always zero.



 - projection  on the direction of the contour ℓ.

We take into account that the work on a closed path is zero

 -circulation of intensity .

Circulation of the electrostatic field, taken by an arbitrary closed path is always zero.

§ 7 Potential.

The link between the intensity and potential.

Gradient of potencial

Equipotential surfaces

Since the electrostatic field is a potential job of moving the charge in such a field can be represented as a difference of the potential energy of a charge in the start and end points of the path. (The work is equal to the reduction of the potential energy, or the change in the potential energy take with minus sign.)


work E


Constant determined from the condition that the removal of the charge q0 to infinity the potential energy must be equal to zero.



Various test charges q0i, placed at a given point of the field will have at this point various potential energies:



The ratio Wpot i to the value of the test charge q0i, placed at a given point of the field is constant for a given point of the field for all test charges. This ratio is called the potential.
Potential - energy characteristic of the electric field. Potential numerically equal to the potential energy, which has at a given point of the field unit positive charge.




Работу по перемещению заряда можно представить в виде



Potential is measured in volts

Potential is measured in volts

Equipotential surface is a surface of equal potential (φ = const). Work to move a charge along an equipotential surface is zero.

 Relationship between the intensity and the potential φ can be found, based on the fact that the job of moving a charge q at the elementary segment d can be written as


On the other hand                          

 - gradient of potential .

Field intensity is equal to the potential gradient, to the negative.

Potential gradient shows how change the capacity per unit length. Gradient perpendicular to the function and is directed towards the increase of the function. Therefore, the vector intensity perpendicular to the equipotential surface and is directed towards the decrease of the potential.

Consider the field created by a system N point charges q1, q2, … qN. Distance from the charge to a given point of the field are r1, r2, … rN. The work done by this field on the charge q0, will be equal to the algebraic sum of the work force each charge separately.




The potential field generated by a system of charges is defined as the algebraic sum of the potentials produced at the same point each charge separately.

§ 9 of difference potentials of the plane, the two planes, spheres, ball, cylinder

Using the relation between φ and  оdefine the potential difference between two arbitrary points


  1.  The potential difference of the field of a uniformly charged infinite plane with surface charge density σ .

2.  The potential difference of the field of two infinite parallel planes with an oppositely charged surface charge density σ .


If х1 = 0; х2 = d , then  or



3.  The potential difference of the field of a uniformly charged spherical surface of radius R .

If r1 = r, r2 , the potential outside the spheres

Inside a spherical surface potential everywhere and is equal


4. The potential difference of the field volume of a charged sphere of radius R and total charge Q .

Outside the ball  r1, r2 > R,

Inside the ball



5. The potential difference of the field of a uniformly charged cylinder (or infinitely long thread).


r > R:




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