that is, the voltage between the poles of the source

current depends on the EMF and the work of external forces to move a unit charge from one pole of the source to the other.

2. Formulate and write Ohm’s law for a closed circuit

Current strength in a closed circuit electrical circuit is proportional to the source emf and inversely proportional to the circuit resistance.

3. What is the difference between counter and matched connection of series-connected current sources?

They say that the 2nd source is switched on counter to the first if they, working alone, create currents flowing in the same direction. The 3rd source is switched on in concert with the first if the currents created by them are directed in the same way.

4. Formulate Ohm’s law for a closed circuit with several current sources connected in series. Give the formula for this law.

The current strength in a closed electrical circuit with current sources connected in series is directly proportional to their sum

EMF is also inversely proportional to the circuit resistance.

5. How to determine the direction of current in a closed circuit with several current sources connected in series?

If

then the current flows clockwise. Otherwise, go counterclockwise.

If points 1 and 2 coincide, then the expression for Ohm’s law for the section takes on a simpler form:

where is the total resistance of the closed circuit including internal resistance sources, and - the algebraic sum of emf. in this chain.

The current that occurs when the external resistance is zero is called short circuit current.

Lecture 10.

Connection of conductors.

Using Ohm's law for a section of a circuit, it can be shown that the resistances of series and parallel connections of conductors are equal, respectively:

Proof:

Note that when connecting conductors in parallel, the total resistance is always less than the smallest resistance in the parallel connection. See for yourself.

Joule-Lenz law.

When current passes through a conductor, resistance generates heat, which is dissipated in the environment. Let's find this amount of heat. For this we will use the law of conservation of energy and Ohm's law.

Let's consider homogeneous section of the circuit where a constant potential difference is maintained. The electric field does work:

If there is no transformation into mechanical, chemical or other types of energy other than thermal in the area, then the amount of heat released is equal to the work of the electric field:

.

The thermal power is equal to:

The final amount of heat is found by integration over time:

This formula expresses the Joule–Lenz law. The mechanism of heat release is associated with the conversion of additional kinetic energy, which current carriers acquire in an electric field, into excitation energy of lattice vibrations when carriers collide with atoms at lattice sites.

Let us find an expression for the Joule–Lenz law in local form. For this purpose, we select an elementary volume in the conductor in the form of a cylinder with a generatrix along the vector. Let the cross section of the cylinder be , and its length . Then, according to the Joule–Lenz law, the amount of heat released in this volume over time is:

where is the volume of the cylinder. Dividing the last ratio by we obtain a formula that determines the thermal power released per unit volume of the conductor:

Specific thermal power is measured in .

The resulting relationship expresses the Joule–Lenz law in local form: the specific thermal power of the current is proportional to the square of the current density and the specific resistance of the conductor at a given point.

In this form, the Joule–Lenz law is applicable to inhomogeneous conductors of any shape, and does not depend on the nature of external forces. If only electrical forces act on the carriers, then based on Ohm’s law:

If a section of the circuit contains an emf source, then the current carriers will be acted upon not only by electrical forces, but also by external forces. In this case, the heat that is released in the area is equal to the algebraic sum of the work of electrical and external forces.

Let's multiply Ohm's law in integral form by the current strength:

Here on the left is (thermal power), and on the right is the algebraic sum of the powers of electrical and external forces, which is called current power.

In a closed circuit:

those. The power of heat generation is equal to the power of external forces.

Ohm's differential law

IN

Let's select a conductor from the array (through which electric current flows I) a small cylinder located along the electric current lines in the conductor Fig. 5.2. Let the length of the cylinder be dl and the cross section dS. Then

ABOUT

here

AND

Using the definition for the current density (5.1) and for the conductivity of the conductor (5.4), we finally obtain the expression, which is called Ohm’s differential law

Work and power produced by electric current

When a charge moves between points with a certain potential difference corresponding to the voltage drop U work and power produced:

E

This law was obtained experimentally and was called the Joule–Lenz law. If, like the previous case, we proceed to the consideration of small volumes, then it is not difficult to obtain the Joule–Lenz law in differential form (5.6-5.8):

Kirchhoff's laws

Kirchhoff's first rule

Let's consider an electrical circuit with branches Fig. 5.3. We will call branching points nodes. In a steady-state process, when the electric current flowing through the circuit is constant, the potentials of all points in the circuit are also unchanged. This can happen if electric charges do not accumulate or disappear at the nodes of the chain.

Thus, in a steady state, the amount of electricity flowing into the node is equal to the amount of electricity leaving the node. It follows from this Kirchhoff's first rule:

The algebraic sum of the forces of electric currents converging at a node is equal to zero (5.9) (currents entering the node are taken with + signs, and currents leaving the node with - sign)

I1+i2+i3-i4-i5=0

ΣI i =0 5.9.

Conductor connections

In practice, it is often necessary to use different connections of conductors

P serial connection Fig.5.4.

P

With such a connection, the electric current in all parts of the circuit and on all its elements is the same I= I 1 = I 2 = I 3 =… I n. The voltage at the ends of the circuit between points A and B is the sum of the voltages at each of its elements U AB = U 1 + U 2 + U 3 +… U n. Thus.

Parallel connection Fig.5.5

Ohm's law for a closed circuit containing e.m.f.

R Let's consider an unbranched electrical circuit containing E.M.F.( E) with internal resistance r and containing external resistance R Fig.5.6

The total work to move the charge along the entire circuit will be the sum of the work in the external circuit and the work inside the source A=A external +A source .

Moreover, the work in the external circuit related to the amount of charge is, by definition, the potential difference on the external circuit (voltage drop on the external circuit) A external / q= U. And the work throughout the circuit related to the charge is, by definition, E.M.F. A/ q= E. From here E= U+ A source / q. On the other side A source = I2 rt. From here A source / q= Ir. Thus we finally get: E= U+ Ir

Or E= I(R+ r) 5.12

Under E implies the sum of all E.M.F. included in an unbranched circuit, and by r and R we mean the sum of all internal and external resistances in the unbranched circuit.

The current strength is the same for the entire unbranched closed circuit containing E.M.F. is directly proportional to the E.M.F. and is inversely proportional to the circuit impedance.

Kirchhoff's second rule

Consider the branched chain Fig. 5.7. Let's call the section between two neighboring nodes a branch. Since branching takes place only at neighboring nodes, within the branch the current strength is maintained in magnitude and direction. Any circuit can be considered as a set of circuits, and for each circuit the following is true:

In any closed circuit, mentally isolated from an electrical circuit, the algebraic sum of the products of the resistances of the corresponding sections of the circuit, including the internal resistances of the sources, and the current strength in the circuit is equal to the algeboraic sum of all E.M.F. in a chain

Ohm's law for a closed circuit

If an electric field is created in a conductor and measures are not taken to maintain it, then the movement of charges will very quickly lead to the fact that the field inside the conductor will disappear and the current will stop, therefore, to maintain direct current For a long time, two conditions must be met: the electrical circuit must be closed; in the electrical circuit along with areas in which the positive

Since the charges move in the direction of decreasing potential, there must be sections in which these charges move in the direction of increasing potential, i.e., against the forces of the electrostatic field (see the part of the circuit depicted by the dashed line in Fig. 5).

Only forces of non-electrostatic origin, called external forces, can move positive charges against the forces of an electrostatic field. A quantity equal to the work of external forces to move a unit positive charge is called electromotive force (EMF) e, acting in a circuit or on its section. EMF e measured in volts (V). The EMF source has some internal resistance, depending on its design. This resistance is connected in series with the source in a common electrical circuit. Galvanic cells and direct current generators are used as sources of EMF (Fig. 6).

If an unbranched closed electrical circuit (Fig. 7) contains several series-connected elements with resistance and sources of emf e to, having internal resistance, then it can be replaced by the equivalent circuit shown in Fig. 6. The current strength in an equivalent circuit is determined by Ohm’s law for a closed circuit:

;

EMF, like current strength, is an algebraic quantity. If the EMF promotes the movement of positive charges in the chosen direction, then e> 0, if the emf prevents the movement of positive charges in a given direction, then e < 0. Чтобы определить знак ЭДС, необходимо показать в электрической цепи направление движения положительных зарядов. Положительные заряды в электрической цепи движутся от положительного полюса источника к отрицательному полюсу. Если по ходу этого направления перейти внутри источника от отрицательного полюса к положительному, то e> 0, if we move inside the source from the positive pole to the negative, then e < 0.

Rice. 6 Fig. 7

From Ohm's law for a closed circuit it follows that the voltage drop U at the source terminals is less than the EMF. Really, e, or e. Since, according to Ohm’s law, for a homogeneous section of the circuit, the voltage at the source terminals is , then

3) using Ohm’s law for a closed circuit, establish the relationship between current strength and EMF.

Tell me Ohm's law

Ohm's law is a physical law that defines the relationship between voltage, current and conductor resistance in an electrical circuit. Named after its discoverer, Georg Ohm.
It so happened that in this section of the page there were two verbal formulations of Ohm’s law:
1. The essence of the law is simple: if, during the passage of current, the voltage and properties of the conductor do not change, then
The current strength in a conductor is directly proportional to the voltage between the ends of the conductor and inversely proportional to the resistance of the conductor.
2. Ohm's law is formulated as follows: The current strength in a homogeneous section of the circuit is directly proportional to the voltage applied to the section, and inversely proportional to the characteristic of the section, which is called the electrical resistance of this section.
It should also be borne in mind that Ohm's law is fundamental and can be applied to any physical system, in which there are flows of particles or fields that overcome resistance. It can be used to calculate hydraulic, pneumatic, magnetic, electrical, light, heat flows, etc., just like Kirchhoff’s Rules, however, this application of this law is used extremely rarely within the framework of highly specialized calculations.

User deleted

The German physicist G. Ohm experimentally established in 1826 that the current strength I flowing through a homogeneous metal conductor (i.e., a conductor in which no external forces act) is proportional to the voltage U at the ends of the conductor:

where R = const.
The value R is usually called electrical resistance. A conductor that has electrical resistance is called a resistor. This relationship expresses Ohm's law for a homogeneous section of the circuit: the current strength in the conductor is directly proportional to the applied voltage and inversely proportional to the resistance of the conductor.
The SI unit of electrical resistance of conductors is the ohm (Ω). A resistance of 1 ohm has a section of the circuit in which a current of 1 A occurs at a voltage of 1 V.
Conductors that obey Ohm's law are called linear. The graphical dependence of the current I on the voltage U (such graphs are called current-voltage characteristics, abbreviated as VAC) is depicted by a straight line passing through the origin of coordinates. It should be noted that there are many materials and devices that do not obey Ohm's law, for example, a semiconductor diode or a gas-discharge lamp. Even for metal conductors, at sufficiently high currents, a deviation from Ohm’s linear law is observed, since the electrical resistance of metal conductors increases with increasing temperature.
For a section of a circuit containing an emf, Ohm's law is written in the following form:
IR = U12 = φ1 – φ2 + E = Δφ12 + E.
This relationship is usually called the generalized Ohm's law.
In this fig. shows a closed DC circuit. The chain section (cd) is uniform.

According to Ohm's law,
IR = Δφcd.
Section (ab) contains a current source with an emf equal to E.
According to Ohm's law for a heterogeneous area,
Ir = Δφab + E.
Adding both equalities, we get:
I(R + r) = Δφcd + Δφab + E.
But Δφcd = Δφba = – Δφab.
That's why

This formula expresses Ohm's law for a complete circuit: the current strength in a complete circuit is equal to the electromotive force of the source divided by the sum of the resistances of the homogeneous and inhomogeneous sections of the circuit.

Little prince

In integral form: i=L*U | L-electrical conductivity, 1/R
In differential form: j=A*E | A - electrical conductivity of the medium, j - current density
For a closed loop: i= E/(r+R) | already brought...
For alternating currents: uo=io*sqrt (r^2 + (w*L -1/w*C)^2) |uo io - amplitudes of current and voltage, r - active resistance of the circuit, which is in brackets and squared - reactive component, sqrt = square root....

Olya Semyonova

Ohm's law is an empirical physical law that determines the relationship between the electromotive force of a source (or electrical voltage) with the strength of the current flowing in the conductor and the resistance of the conductor. Installed by Georg Ohm in 1826 and named after him.

Anyone who has chosen repair and maintenance of electrical installations as their specialty is well aware of the statement of teachers: “You need to know Ohm’s law for a closed circuit. Even waking up in the middle of the night, it is important to be able to formulate it. Because this is the basis of all electrical engineering.” Indeed, the pattern discovered by the outstanding German physicist Georg Simon Ohm influenced the subsequent development of the science of electricity.

In 1826, while conducting experiments to study the passage through a conductor, Ohm revealed a direct relationship between the voltage of the power source supplied to the circuit (although in this case it is more correct to talk about the electromotive force of the EMF) and the resistance of the conductor itself. The dependence was theoretically substantiated, resulting in Ohm's law for a closed circuit. An important feature: the relevance of the identified fundamental law is valid only in the absence of an external disturbing force. In other words, if, for example, a conductor is in an alternating magnetic field, then direct application of the formulation is impossible.

Ohm's law for a closed circuit was revealed during the study the simplest scheme: a power source (having an EMF), conductors go from its two terminals to the resistor, in which the directed movement of charge-carrying elementary particles occurs. Hence, the current is the ratio of the electromotive force to the total resistance of the circuit:

where E is the electromotive force measured in volts; I - current value, in amperes; R is the electrical resistance of the resistor, in Ohms. Note that Ohm's law for a closed circuit takes into account all components of R. When calculating a complete closed circuit, R is understood as the sum of the resistances of the resistor, conductor (r), and power source (r0). That is:

If the source r0 is greater than the sum of R+r, then the current strength does not depend on the characteristics of the connected load. In other words, the source of EMF in this case is If the value of r0 is less than R+r, then the current is inversely proportional to the total external resistance, and the power source generates voltage.

By doing accurate calculations even voltage loss at connections is taken into account. The electromotive force is determined by measuring the potential difference at the source terminals when the load is off (the circuit is open).

Ohm's laws apply to a section of a circuit just as often as to a closed loop. The difference is that the calculations do not take into account the EMF, but only the potential difference. Such an area is called homogeneous. In this case, there is a special case that allows you to calculate the characteristics on each of its elements. Let's write it in the form of a formula:

where U is voltage or potential difference, in volts. It is measured with a voltmeter by connecting probes in parallel to the terminals of any element (resistance). The resulting value of U is always less than the EMF.

Actually, this formula is the most famous. Knowing any two components, you can find the third from the formula. The calculation of contours and elements is performed using the law under consideration for a section of the circuit.

Ohm's law for a magnetic circuit is in many ways similar to its interpretation for an electrical circuit. Instead of a conductor, a closed magnetic circuit is used; the source is the winding of a coil with current passing through the turns. Accordingly, the emerging one is closed along the magnetic circuit. The magnetic flux (F) circulating along the circuit directly depends on the value of the MMF (magnetomotive force) and the resistance of the material to the passage of the magnetic flux:

where F is the magnetic flux, in Webers; F - MMF, in amperes (sometimes gilberts); Rm is the resistance causing attenuation.

A complete closed circuit (Fig. 1) can be considered as a series connection of the external circuit resistance (R) and the internal resistance of the current source (r). That is:

If you replace the current source with such that its internal resistance is equal to the same resistance as the previous one, then the current in the circuit will change. That is, the current in the circuit depends on both the internal resistance of the source and its EMF. Quantitatively, all these quantities: EMF ($\mathcal E$) of the source, its internal resistance, current strength in the circuit (I), electrical resistance of the circuit (R) are related by Ohm’s law.

Relationship between Ohm's local law and the integral law for a closed circuit

Let's assume that electric currents flow in thin wires. In this case, the directions of the currents coincide with the direction of the wire axis. For thin wires, we can assume that the current density is $\overrightarrow(j)=const$ at any point in the cross-section of the wire. In our case, we can write that the current strength is equal to:

where $S$ is the cross-sectional area of ​​the conductor. Let us deal with direct current (I=const) along the entire conductor. Let's assume that there is an EMF source ($\mathcal E$) in the circuit. In this case, the local formulation of Ohm's law will look like:

where $\overrightarrow(E)$ is the field strength of Coulomb forces, $\overrightarrow(E_(stor))$ is the field strength of external forces, $\sigma $ is specific conductivity, $\overrightarrow(e)$ is the unit vector, current-directed. For a thin wire, expression (3) can be written as:

Let's multiply expression (4) by the element of the length of the conductor (dl) and find the integral over the section of the conductor from point 1 to point 2. Since we assumed the current strength to be constant, we have:

The electrostatic field is potential, therefore:

The second integral in expression (5) is not equal to zero only within the limits of the EMF source. It does not depend on the position of points 1 and 2. They must only be located outside the source.

The emf of the source is considered to be greater than zero if path 1-2 crosses the source from the negative pole to the positive pole.

where $R"$ is electrical resistance, $\rho$ -- resistivity. Thus, from expression (5) we obtain:

We have obtained Ohm's law in integral form. If the circuit is closed, then $(\varphi )_1=(\varphi )_2$, therefore:

where $R"$ is the electrical resistance of the entire circuit, the electrical resistance of the load and the internal resistance of the current source. That is, we write Ohm’s law for a closed circuit as:

where $r$ is the electrical resistance of the current source.

Quite often we have to solve problems in which the voltage at the ends of a section of the circuit is not known, but the resistances are given components circuit and the emf of the source that powers the circuit. Then use Ohm's law in the form (11) to calculate the current strength that flows in the circuit.

Example 1

Task: The current source has an internal electrical resistance equal to r. Find the potential drop inside the source ($U_r$) inside the element if the current in the circuit is I. How to calculate the external electrical resistance of the circuit under given conditions?

As a basis for solving the problem, we use Ohm’s law for a closed circuit:

From formula (1.1) it is easy to obtain a formula for calculating external resistance:

In order to calculate the voltage drop inside the current source, we use Ohm's law for the circuit section:

\[(I=\frac(U_r)(r)\to U)_r=Ir\ \left(1.2\right).\]

Answer: $U_r=Ir,$ $R=\frac(\mathcal E)(I)-r.$

Example 2

Assignment: The current source has an internal resistance equal to r=1 Ohm and an emf equal to $\mathcal E$=10V. Find the source efficiency ($\eta $) if the current in the circuit is I=5 A.

The efficiency of the current source is equal to the ratio:

\[\eta =\frac(P")(P)\left(2.1\right),\]

where $P"$ is the power (useful power) that is released by the external section of the circuit, $P$ is the total power that is developed by the source. In this case:

\ \

Therefore, the source efficiency can be expressed as:

\[\eta =\frac(I^2R\ )(\mathcal E I)=\frac(IR)(\mathcal E)\left(2.4\right).\]

Following Ohm's law for a closed circuit we write:

Let us express from (2.5) the electrical resistance of the external circuit, we obtain:

Substituting (2.6) into the expression for efficiency (2.4), we obtain:

\[\eta =\frac(I\left(\frac(\mathcal E)(I)-r\right))(\mathcal E)=\frac(\mathcal E-Ir)(\mathcal E).\ ]

Let's substitute the numerical data, carry out calculations, and get:

\[\eta =\frac(10-5\cdot 1)(10)\cdot 100\%=50\%\]

In this article we will talk about Ohm’s law, formulas for a complete circuit (closed), a section of a chain, a non-uniform section of a chain, in differential and integral form, alternating current, as well as for a magnetic circuit. You will learn which materials do and do not conform to Ohm's law, and where it occurs.
direct current flowing through a conductor is directly proportional to the voltage , applied to its ends and is inversely proportional to the resistance.

Ohm's law was formulated by the German physicist and mathematician Georg Ohm in 1825-26 based on experience. This is an experimental law, not a universal one - it applies to some materials and conditions.

Ohm's law is a special case of the later and more general - Kirchhoff's second law

Below is a video that explains Ohm's law on fingers.

Ohm's law formula for a section of a circuit

The intensity of direct current flowing through a conductor is proportional to the voltage applied to its ends. On the Internet this formula is often called Ohm’s first law:

U- voltage

I- current strength (intensity)

R - Resistance

Electrical resistance:

The proportionality factor R is called electrical resistance or resistance.

The ratio of voltage to current for a given conductor is constant:

The unit of electrical resistance is 1 ohm (1 Ω):

A resistor has a resistance of 1 if the applied voltage is 1 volt and the current is 1 ampere.

Dependence of electrical resistance on guide size:

The resistance of a conducting section with a constant cross-section R is directly proportional to the length of this segment li, inversely proportional to the cross-sectional area S:

R- electrical resistance

ρ - resistivity

I— guide length

S- cross-sectional area

This relationship was confirmed experimentally by British physicist Humphrey Dee in 1822 before the development of Ohm's law.

Ohm's law for a closed (complete) circuit

- this is the value of the strength (intensity) of the current in a real circuit, which depends on the load resistance and on the current source (E), also called Ohm’s second law.

An electric light bulb is a consumer of a current source; when connected together, they create a complete electrical circuit. In the picture above, you can see a complete electrical circuit consisting of a battery and an incandescent lamp.

Electricity passes through the incandescent lamp and through the battery itself. Consequently, the current passing through the lamp will subsequently pass through the battery, that is, the resistance of the lamp is added to the resistance of the battery.

The load resistance (light bulb) is called external resistance , and the resistance of the current source (battery) is internal resistance . Battery resistance is designated by the Latin letter r.

When electricity flows around a circuit, the internal resistance of the cell itself resists the flow of current, and therefore thermal energy is lost within the cell itself.

• E = electromotive force in volts, V
• I = current in amperes, A
• R = circuit load resistance in Ohms, Ω
• r = internal cell resistance in Ohms, Ω

We can change this equation;

In this equation appears ( V), That is finite potential difference, measured in volts (V). This is the potential difference at the cell terminals when current flows in the circuit; it is always less than the emf. cells.

Ohm's law for a non-uniform section of a circuit

If only potential forces act on a section of the chain ( Figure 1a), then Ohm's law is written in a known form. If the action of external forces also appears in the circle ( Figure 2b), then Ohm's law will take the form , where . This is Ohm's law for any section of the circuit.

Ohm's law can be extended to the entire circle. By connecting points 2 and 1 ( Figure 3c), we transform the potential difference to zero, and taking into account the resistance of the current source, Ohm's law will take the form . This is the expression of Ohm's law for a complete circuit.

The last expression can be represented in various forms. As is known, the voltage in the external section depends on the load, that is
or , or .

In these expressions Ir is the voltage drop inside the current source, and it can also be seen that the voltage U less than ε by the amount Ir. Moreover, the greater the external resistance compared to the internal, the greater U approaches ε.

Let's consider two special cases regarding the external resistance of the circuit.

1) R = 0 - this phenomenon is called a short circuit. Then, from Ohm's law we have - , that is, the current in the circuit increases to a maximum, and the external voltage drop U→ 0. In this case, the source is highlighted high power, which may cause it to malfunction.

2) R= ∞ , that is, the electrical circuit is broken, then , A . So, in this case, the EMF is numerically equal to the voltage at the terminals of the open current source.

Ohm's law in differential form

Ohm's law can be presented in such a way that it is not related to the size of the conductor. Let's select a section of the conductor Δ l, at the ends of which potentials φ 1 and φ 2 are applied. When the average cross-sectional area of ​​the conductor Δ S, and the current density j, then the current strength

If Δ l → 0, then taking the limit of the relationship, . So, we finally get , or in vector form - this is the expression Ohm's law in differential form. This law expresses the current strength at an arbitrary point of a conductor depending on its properties and electrical state.

Ohm's law for alternating current

This equation is written Ohm's law for alternating current circuits relative to their amplitude values. It is clear that it will also be valid for effective values ​​of force and current: .

For AC circuits, a case is possible when , which means that UL = UC. Since these voltages are in antiphase, they compensate each other. Such conditions are called voltage resonance. Resonance can be achieved either at ω = const, changing WITH And L, or at constant WITH And L select ω, which is called resonant. As seen - .

Features of voltage resonance are as follows:

Finally, from (2) - (4) we have expression for Ohm's law in integral form

which he established experimentally.

Interpretation of Ohm's law

The current intensity, which is the effect of the applied voltage, behaves proportionally to its voltage. For example: if the applied voltage doubles, it also doubles the current (current intensity).

Remember that Ohm's law is only satisfied by some materials - mainly metals and ceramic materials.

When does Ohm's law occur and what materials do and do not conform to Ohm's law

Ohm's law is an experimental law fulfilled for certain materials (such as metals) for fixed current conditions, in particular the temperature of the conductor.

Materials that fall under Ohm's law are called ohmic guides or linear conductors. Examples of conductors that follow Ohm's law include metals (eg, copper, gold, iron), some ceramics, and electrolytes.

Non-Ohm's Law materials, in which resistance is a function of the intensity of the current flowing through them, are called non-linear conductors. Examples of non-Ohm's Law guidelines include semiconductors and gases.

Ohm's law does not hold true when conductor parameters, especially temperature, change.