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Electric Field Lines Caused by Point Charges

 *

that, d E t


due to dQ * symmetrical to d E t about the OM axis , this electric field pair

pairwise cancel each other out so the sum of the d E t components over the entire conductor is zero. What remains is:

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E M d En v loop


Because the vectors d E n have the same direction, E M has a point at M , has the direction of the loop axis and points away from the loop. In terms of magnitude:

E M

round​

dE n


Step 4: Find the relevant quantities and solve. To find the solution to the integration problem, it is necessary to convert all the variables into a single variable.

According to the figure, we have the force caused by dQ at M equal to:

dE n dE cos

( is the angle between d E and OM ). Electric field

dE k dQ

r 2

Because: cos h ,

r


r 2 R 2 h 2


should:

dE n


dE n

k hdQ

r 3

k hdQ


3

( R 2 h 2 ) 2


hQ 3

So:

E M

round​

dE n k

( R 2 h 2 )

2 dQ

round​



Comment:

E M k

hQ ( R 2 h 2 ) 2

3


At the center of the loop: h0, therefore E 0 0


At a place quite far from the loop: h R , r h , E M

kQ

h 2

If the wire loop is negatively charged ( Q 0 ) then EM tends towards the center O of the wire loop and has magnitude.


E M k

hQ ( R 2 h 2 ) 2

3


1.4. ELECTRICITY


1.4.1. Electric field lines


a. Definition

Electric field lines are curves whose tangent at each point coincides with the direction of the electric field intensity vector at that point, the direction of the electric field lines is the direction of the electric field intensity vector.

Thus, the lines of force of a uniform electric field are parallel lines, the lines of force of the electric field caused by a point charge are straight lines directed towards the point charge if it is negative and directed towards the

away from the point charge if it is positive.

b. Convention


Figure 1.4. Electric field lines


Figure 1.5. Electric field lines caused by a point charge


The number of electric field lines passing through a unit area perpendicular to the field lines is proportional to the electric field intensity E at the place where that area is located.

The electric field lines are well distributed .​​​​​

call

is the spectrum of electric field or electric current


Number of electric lines of force

~ E

Area


(1.14)


From the above equation , through the spectrum, if the density of the electric current is large ( thick ) ,

where the electric field density is small ( sparse ) , there is a weak electric field . With a uniform electric field ( E ​​​const ) electric field is a series of parallel lines that are equidistant from each other.

Comment

The electric field originates from a positive charge and terminates on a negative charge .​​

The curves of the magnetic field are similar to those of the magnetic field .​​​​​​

The electric field lines do not intersect because at each point in the electric field vector E there is only one definite direction .

1.4.2. Interruption

electric field vector - Electric induction vector​​​​​​


From the expression of electric field strength E , it can be seen that in the environment, the electric field strength is reduced by many times compared to that in a vacuum.

Thus, when passing through the interface between vacuum and the environment, the field line spectrum will be interrupted at the interface.

For convenience of calculation, we remove that discontinuity by introducing a quantity called the electric induction vector D defined as:



Figure 1.6. Discontinuity of electric field lines

D  0 E (1.15)

So the electric induction vector caused by a point charge q at a point M

q at an interval r is:


1

q

r



D (1.16)

4 r 2 r


q

D ( C m 2 ) (1.17)

4 r 2

At each point in the electric field, D depends only on the source generating the electric field and does not depend on the properties of the medium.

The concept of lines of force is given corresponding to the vector quantity of electric field intensity, similarly, people give the concept of electric induction lines corresponding to the electric induction vector as follows:

An electric induction line is a curve whose tangent at each point coincides with the direction of the electric induction vector, the direction is the direction of the electric induction vector. The number of electric induction lines passing through a unit area placed perpendicularly is proportional to the magnitude.

electric induction vector at the location of the area under consideration.


Number of touch lines

~ D

Area


(1.18)

Figure 1.7. Continuity of electric induction lines


Thus, when passing through the interface between two media with different dielectric constants, the electric induction lines are continuous while the field lines are interrupted.

1.4.3. Electricity


a. Definition

The electric flux sent through an area S is a quantity whose value is proportional to the number of electric induction lines that intersect that area.

b. Expression

First we consider the electric flux of a uniform electric field sent through a flat cross section.

S n

Comment:The number of electric induction lines cutting any flat cross-section S placed in electric field D is equal to the number of electric induction lines


28 Figure 1.8. Electric flux of electricity

uniform field sent through plane section

intersect S n is the projection of S on the plane perpendicular to D .

Let n be the normal vector of the cross-section S , n makes an angle  with D. We have: S n S . cos .

Symbol: S S . n is called the area vector S.

From the definition of electric flux and convention (1.18) we have: electric flux sent through area S is equal to electric flux sent through S n and is equal to:

e D . S n D . S . cos D n . S D . S (1.19)


Attention:

0

2e

0

2e

0

2e


Consider the electric flux of any electric field sent through a cross-section of arbitrary shape.


Figure 1.9. Electric flux sent through area dS

Then, it is easy to convert to the case of uniform electric field and flat cross-section by dividing the surface S into countless elements of infinitely small area dS so that through each element dS can be considered flat and the electric field sent through dS is uniform.

The electric flux d e of D sent through dS is:

d e D . dS D . dS . cos( D , dS ) (1.20) The electric flux sent through the entire area S is:

e d e D . dS D n . dS

(1.21)


Attention

( S )

( S )

( S )

For a closed surface, we always choose the direction of the normal n to be the direction outwards of that closed surface.

If D is directed outwards from the closed surface e 0 (for example, the electric flux sent through the sphere surrounding the positive charge receives a positive value).

If D is directed towards the closed surface e 0 (for example, the electric flux sent through the sphere surrounding the negative charge receives a negative value).


1.5. OXTROGRATXKI - GAUSS THEOREM (O - G) FOR ELECTRIC FIELD

1.5.1. Establishing the theorem


Calculate the electric flux of a point charge sent through a closed surface in the following cases:

after:


+

0

The closed surface S 0 is a sphere surrounding a point charge whose center coincides with the point charge.

e ( S 0 )

( S 0 )

D . dS


1

q

r



in which: D

4 r 2 r

  r

Figure 1.10. Electric flux through a closed surface S 0

dS dS . n dS .

r


Comment:At every point on the sphere, D has the same magnitude and is directed along the radius of the sphere, meaning D is in the same direction as dS . So:

( S )

dS q ( r ) 2

dSq q

dS q

4 r 2

e 0

( S 0 )

4 r 2 r


( S 0 )

4 r 2

4 r 2


( S 0 )

4 r 2



Comment:

e ( S 0 ) q . (1.22)

The electric flux sent by a point charge through a sphere does not depend on the radius of the sphere.

There are q electric induction lines going out of the sphere from a positive point charge and conversely there are q electric induction lines going into the sphere from a negative charge.

A closed surface is any surface that surrounds a point charge.

Consider a closed surface S 1 which is any surface surrounding charge q in which the electric induction lines intersect S 1 only once .

Figure 1.11

Because the electric induction lines are continuous even at the interface between two different media, there are as many electric induction lines cutting S 0 as there are electric induction lines cutting S 1 .

By definition, the electric flux is proportional to the number of electric induction paths, so the electric flux sent through two areas S 0 and S 1 is equal.

( S ) ( S ) q .

e 0 e 1


Consider a closed surface S 2 as any surface surrounding charge q. Then the electric induction lines can only intersect S 2 an odd number of times.

Consider the case where the electric induction line intersects S 2 3 times (and q 0 ), we have:

First time, the electric induction line goes out of the closed surfacee1 ,

The second time, the electric induction line goes into the closed surfacee1 ,

Third time, the electric induction line goes out of the closed surfacee1 .

Thus, even if the induction line cuts the closed surface an odd number of times , it will only have the same effect in terms of electrical flux as the induction line cutting the closed surface once. That is, the electric flux sent through S 1 and S 2 is the same:

e ( S 0 ) e ( S 1 ) e ( S 2 ) q .

Closed surface S 3 does not surround the charge (charge lies outside the closed surface)

If the charges are not inside the closed surface, the electric flux sent through that closed surface is zero because the number of induction paths entering is equal to the number of induction paths leaving that closed surface.


Conclude

e ( S 3 ) 0


The electric flux caused by a charge sent through a closed surface has the value q if q is inside the closed surface and is zero if q is outside the closed surface.

Electric flux does not depend on the position of the charge in the closed surface and does not depend on the choice of shape of the closed surface.


Figure 1.12

In case there are many charges, the electric flux sent through the closed surface is equal to the sum of the electric fluxes caused by each charge on that closed surface.

1.5.2. Theorem statement


The electric flux sent through a closed surface is equal to the algebraic sum of the charges contained in that closed surface:

e( S ) D . dS q i

(1.23)

( S ) i


1.5.3. Differential form of the OG theorem


According to vector analysis:



DivD


dD x dx


dD y

dy


dDz​

dz

  O G

DdS  divD . dV q i

( S ) ( V ) i


In the case of continuous charge distribution with mass density , we have:

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