Lecture 2.6.
Charge interaction energy
Consider a system of two point charges. The interaction energy can be interpreted as the energy of the first charge in the field of the second (see (2.1.3))
Since both representations are equal, the interaction energy of these charges can be written as follows
Where - i-th point charge of the system, is the potential of the field created by all other charges of the system, except i-that, at the point where the charge is located.
If the charges are distributed continuously, then, representing the system of charges as a collection of elementary charges and proceeding to integration, we obtain the expression
where is the energy of interaction of the elementary charges of the first ball with each other, is the energy of interaction of the elementary charges of the second ball with each other, is the energy of interaction of the elementary charges of the first ball with the elementary charges of the second ball. Energy is called own energies charges and . Energy is called interaction energy charges and .
Energy of an isolated conductor and capacitor
Let the conductor have charge and potential. Conductor energy. Since the conductor is an equipotential region, the potential is taken out from under the integral sign. Finally
Capacitor energy.
Let and be the charge and potential of the positively charged plate, and and be the negative plate, respectively. Then the energy of the capacitor, taking into account and will be written
Electric field energy.
The physical meaning of the energy of a capacitor is nothing more than the energy of the electric field concentrated inside it. Let us obtain an expression for the energy of a flat capacitor in terms of voltage. We will neglect edge effects. Let's use the formula and expression for the capacitance of a flat capacitor.
The integrand here has the meaning of energy contained in the volume. This leads to an important idea about localization of energy in the field itself.
This assumption is confirmed in the field of variable fields. It is alternating fields that can exist independently of the electrical charges that excite them and propagate in space in the form of electromagnetic waves that transfer energy.
Thus, the carrier of energy is the field itself.
Analyzing the last expression, we can introduce the volumetric energy density, i.e. energy contained in a unit volume
| . | (2.6.9) |
We obtained (2.6.8) and (2.6.9) in the special case of a homogeneous, isotropic dielectric in a uniform electric field. In this case, the vectors and are codirectional and can be written
Potential energy of interaction of a system of point charges and total electrostatic energy of a system of charges
Animation
Description
The potential energy of interaction between two point charges q 1 and q 2 located in a vacuum at a distance r 12 from each other can be calculated by:
(1)
Consider a system consisting of N point charges: q 1, q 2,..., q n.
The interaction energy of such a system is equal to the sum of the interaction energies of charges taken in pairs:
. (2)
In formula 2, the summation is carried out over the indices i and k (i № k). Both indices range, independently of each other, from 0 to N. Terms for which the value of index i coincides with the value of index k are not taken into account. The coefficient 1/2 is set because when summing, the potential energy of each pair of charges is taken into account twice. Formula (2) can be represented as:
, (3)
where j i is the potential at the point where the i-th charge is located, created by all other charges:
.
The interaction energy of a system of point charges, calculated using formula (3), can be either positive or negative. For example, it is negative for two point charges of opposite sign.
Formula (3) does not determine the total electrostatic energy of a system of point charges, but only their mutual potential energy. Each charge qi, taken separately, has electrical energy. It is called the charge’s own energy and represents the energy of mutual repulsion of infinitely small parts into which it can be mentally broken down. This energy is not taken into account in formula (3). Only the work spent on bringing charges q i closer together is taken into account, but not on their formation.
The total electrostatic energy of a system of point charges also takes into account the work required to form charges q i from infinitely small portions of electricity transferred from infinity. The total electrostatic energy of a system of charges is always positive. This is easy to show using the example of a charged conductor. Considering a charged conductor as a system of point charges and taking into account the same potential value at any point of the conductor, from formula (3) we obtain:
This formula gives the total energy of a charged conductor, which is always positive (for q>0, j>0, therefore W>0, if q<0 , то j <0 , но W>0 ).
Timing characteristics
Initiation time (log to -10 to 3);
Lifetime (log tc from -10 to 15);
Degradation time (log td from -10 to 3);
Time of optimal development (log tk from -7 to 2).
Diagram:
Technical implementations of the effect
Technical implementation of the effect
To observe the interaction energy of a system of charges, it is enough to hang two light conductive balls on strings at a distance of about 5 cm from each other and charge them with a comb. They will deviate, that is, they will increase their potential energy in the field of gravity, which is done due to the energy of their electrostatic interaction.
Applying an effect
The effect is so fundamental that without exaggeration it can be considered that it is applied to any electrical and electronic equipment that uses charge storage devices, that is, capacitors.
Literature
1. Savelyev I.V. Course of general physics. - M.: Nauka, 1988. - T.2. - P.24-25.
2. Sivukhin D.V. General course of physics. - M.: Nauka, 1977. - T.3. Electricity.- P.117-118.
Keywords
- electric charge
- point charge
- potential
- potential interaction energy
- total electrical energy
Sections of natural sciences:
Superposition principle.
If an electric field created by several charged bodies is studied using a test charge, then the resulting force turns out to be equal to the geometric sum of the forces acting on the test charge from each charged body separately. Consequently, the electric field strength created by a system of charges at a given point in space is equal to the vector sum of the electric field strengths created at the same point by charges separately:
This property of the electric field means that the field obeys superposition principle. In accordance with Coulomb's law, the strength of the electrostatic field created by a point charge Q at a distance r from it is equal in magnitude:
This field is called Coulomb field. In a Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: if Q is greater than 0, then the intensity vector is directed away from the charge, if Q is less than 0, then the intensity vector is directed towards the charge. The magnitude of the tension depends on the size of the charge, the environment in which the charge is located, and decreases with increasing distance.
The electric field strength created by a charged plane near its surface:
So, if the problem requires determining the field strength of a system of charges, then we must proceed according to the following algorithm:
1. Draw a picture.
2. Draw the field strength of each charge separately at the desired point. Remember that tension is directed towards a negative charge and away from a positive charge.
3. Calculate each of the tensions using the appropriate formula.
4. Add the stress vectors geometrically (i.e. vectorially).
Potential energy of interaction of charges.
Electric charges interact with each other and with the electric field. Any interaction is described by potential energy. Potential energy of interaction of two point electric charges calculated by the formula:
Please note that the charges have no modules. For unlike charges, the interaction energy has a negative value. The same formula is valid for the interaction energy of uniformly charged spheres and balls. As usual, in this case the distance r is measured between the centers of the balls or spheres. If there are not two, but more charges, then the energy of their interaction should be calculated as follows: divide the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.
Problems on this topic are solved, like problems on the law of conservation of mechanical energy: first, the initial energy of interaction is found, then the final one. If the problem asks you to find the work done by moving charges, then it will be equal to the difference between the initial and final total energy of interaction of charges. Interaction energy can also be converted into kinetic energy or other types of energy. If the bodies are at a very large distance, then the energy of their interaction is assumed to be equal to 0.
Please note: if the problem requires finding the minimum or maximum distance between bodies (particles) when moving, then this condition will be met at that moment in time when the particles move in one direction at the same speed. Therefore, the solution must begin by writing down the law of conservation of momentum, from which this identical speed is found. And then we should write the law of conservation of energy, taking into account the kinetic energy of particles in the second case.
When a charge is removed to infinity
r2 = ∞ U=U2 = 0,
potential charge energy q2,
charge located in the field q1
at a distance r
17. Potential. Field potential of a point charge.
Potential charge energy q in the field n charges qi
Attitude U/q does not depend on the amount of charge q and is energy characteristics electrostatic field called potential.
The potential at a point in an electrostatic field is a physical quantity numerically equal to the potential energy of a single positive charge placed at this point. This is a scalar quantity.
In SI φ measured in Volts [V = J/C]
1 V is the potential of a point in the field at which a charge of 1 C has an energy of 1 J.
E - [N/C = N m/C m = (J/C) (1/m) = V/m].
Point charge field potential
Potential is a more convenient physical quantity compared to tension E
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Potential energy of a charge in the field of a system of charges. Superposition principle for potentials.
Point charge system: q1,q2, …qn.
The distance from each charge to a certain point in space: r1,r2, …rn.
Work done on the charge q electric field of the remaining charges when it moves from one point to another, is equal to the algebraic sum of the work caused by each of the charges separately
ri 1 – distance from charge qi to the initial charge position q,
ri 2 – distance from charge qi to the final charge position q.
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ri 2 → ∞
Potential difference. Equipotential surfaces
When moving a charge q 0+ in an electrostatic field from point 1 to point 2
r2 = ∞ → U 2 = U∞ = 0
Potential– a physical quantity determined by the work of moving a unit positive charge from a given point to infinity.
When they talk about potential, they mean the potential difference ∆ φ between the point in question and the point, potential φ which is taken as 0.
Potential φ does not have any physical meaning at a given point, since it is impossible to determine the work at a given point.
Equipotential surfaces (surfaces of equal potential)
1) potential at all points φ has the same meaning
2) electric field strength vector E always normal to equipotential surfaces,
3) ∆φ between any two equipotential surfaces is the same
– 
For a point charge
φ = const.
r = const.
For a uniform field, equipotential surfaces are parallel lines.
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The work done to move a charge along an equipotential surface is zero.
because φ 1 = φ 2.
20. Tension vector connection E and potential differences.
Work to move a charge in an electric field:
The potential energy of the electric field depends on the coordinates x, y, z and is a function U(x,y,z).
When moving a charge:
(x+dx), (y+dy), (z+dz).
Change and potential energy:
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From (1)
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Nabla operator (Hamilton operator).
Within electrostatics, it is impossible to answer the question of where the energy of a capacitor is concentrated. The fields and the charges that formed them cannot exist separately. They cannot be separated. However, alternating fields can exist regardless of the charges that excite them (solar radiation, radio waves, ...), and they transfer energy. These facts force us to admit that the carrier of energy is the electrostatic field .
When moving electric charges, the Coulomb interaction forces do a certain amount of work d A. The work done by the system is determined by the decrease in interaction energy -d W charges
| . | (5.5.1) |
Interaction energy of two point charges q 1 and q 2 located at a distance r 12, is numerically equal to the work of moving the charge q 1 in the field of a stationary charge q 2 from point with potential to point with potential:
| . | (5.5.2) |
It is convenient to write down the interaction energy of two charges in a symmetric form
 .
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(5.5.3) |
For a system from n point charges (Fig. 5.14) due to the principle of superposition for potential, at the point of location k-th charge, we can write:
Here φ k , i- potential i-th charge at the location point k-th charge. In total, the potential φ is excluded k , k, i.e. The effect of the charge on itself, which is equal to infinity for a point charge, is not taken into account.
Then the mutual energy of the system n charges is equal to:
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(5.5.4) |
This formula is valid only if the distance between the charges significantly exceeds the size of the charges themselves.
Let's calculate the energy of a charged capacitor. The capacitor consists of two, initially uncharged, plates. We will gradually remove charge d from the bottom plate q and transfer it to the top plate (Fig. 5.15).
As a result, a potential difference will arise between the plates. When transferring each portion of charge, elementary work is performed
Using the definition of capacity we get
Total work expended to increase the charge on the capacitor plates from 0 to q, is equal to:
This energy can also be written as
