Joule's Law - Lenz

The Joule-Lenz law is a physical law that quantifies the thermal effect of an electric current . It was established in 1841 by James Joule and independently of it in 1842 by Emilie Lenz ^[1] .

Content

Definitions

In verbal wording it reads as follows ^[2] :

The power of heat generated per unit volume of the medium during the flow of direct electric current is equal to the product of the electric current density and the electric field strength .

Mathematically it can be expressed in the following form:

w={\vec {j}}\cdot {\vec {E}}=\sigma E^{2},

{\ displaystyle w = {\ vec {j}} \ cdot {\ vec {E}} = \ sigma E ^ {2},}

Where $w$ ${\ displaystyle w}$ - power of heat generation per unit volume, ${\vec {j}}$ ${\ displaystyle {\ vec {j}}}$ - electric current density , ${\vec {E}}$ ${\ displaystyle {\ vec {E}}}$ Is the electric field strength , σ is the conductivity of the medium, and the dot indicates the scalar product.

The law can also be formulated in integral form for the case of currents flowing in thin wires ^[3] :

The amount of heat released per unit time in the considered section of the circuit is proportional to the product of the square of the current strength in this section and the resistance of the section.

In integral form, this law has the form

dQ=I^{2}Rdt,

{\ displaystyle dQ = I ^ {2} Rdt,}

Q=\int \limits _{t_{1}}^{t_{2}}I^{2}Rdt,

{\ displaystyle Q = \ int \ limits _ {t_ {1}} ^ {t_ {2}} I ^ {2} Rdt,}

Where $d Q$ ${\ displaystyle dQ}$ - the amount of heat released over a period of time $d t$ ${\ displaystyle dt}$ , $I$ ${\ displaystyle I}$ - current strength $R$ ${\ displaystyle R}$ - resistance $Q$ ${\ displaystyle Q}$ - the total amount of heat released over a period of time from $t_{1}$ ${\ displaystyle t_ {1}}$ before $t_{2}$ ${\ displaystyle t_ {2}}$ . In the case of constant current and resistance:

Q=I^{2}Rt.

{\ displaystyle Q = I ^ {2} Rt.}

Applying Ohm's law, one can obtain the following equivalent formulas:

Q=U^{2}t/R\ =IUt.

{\ displaystyle Q = U ^ {2} t / R \ = IUt.}

Practical Importance

Energy Loss Reduction

When transmitting electricity, the thermal effect of current in the wires is undesirable, since it leads to energy losses. Lead wires and load are connected in series , meaning mains current $I$ ${\ displaystyle I}$ the wires and the load are the same. The load power and wire resistance should not depend on the choice of source voltage. The power allocated on the wires and on the load is determined by the following formulas

Q_{w}=R_{w}\cdot I^{2},

{\ displaystyle Q_ {w} = R_ {w} \ cdot I ^ {2},}

Q_{c}=U_{c}\cdot I.

{\ displaystyle Q_ {c} = U_ {c} \ cdot I.}

Whence it follows that $Q_{w}=R_{w}\cdot Q_{c}^{2}/U_{c}^{2}$ ${\ displaystyle Q_ {w} = R_ {w} \ cdot Q_ {c} ^ {2} / U_ {c} ^ {2}}$ . Since in each case the load power and resistance of the wires remain unchanged and the expression $R_{w}\cdot Q_{c}^{2}$ ${\ displaystyle R_ {w} \ cdot Q_ {c} ^ {2}}$ is a constant, then the heat generated on the wire is inversely proportional to the square of the voltage at the consumer. By increasing the voltage we reduce the heat loss in the wires. This, however, reduces the electrical safety of power lines .

Wire Selection for Circuits

The heat generated by the conductor with current, to one degree or another, is released into the environment. If the current strength in the selected conductor exceeds a certain maximum permissible value, so much heating is possible that the conductor can provoke a fire in the objects adjacent to it or melt itself. As a rule, when choosing wires intended for the assembly of electrical circuits, it is enough to follow the adopted regulatory documents that regulate the choice of cross-section of conductors.

Electric heaters

If the current strength is the same throughout the entire electrical circuit, then in any selected area it will generate more heat, the higher the resistance of this area.

Due to a conscious increase in the resistance of a circuit section, localized heat generation in this section can be achieved. According to this principle, electric heaters work. They use a heating element - a conductor with high resistance. An increase in resistance is achieved (jointly or separately) by selecting an alloy with a high resistivity (e.g., nichrome , constantan ), increasing the length of the conductor and reducing its cross section. Lead wires have the usual low resistance and therefore their heating is usually invisible.

Fuses

To protect electrical circuits from the flow of excessively high currents, a piece of conductor with special characteristics is used. It is a conductor of relatively small cross section and made of an alloy such that at acceptable currents the heating of the conductor does not overheat it, and with excessively large conductor overheating is so significant that the conductor melts and opens the circuit.

Notes

↑ Joule - Lenza law // Debtor - Eucalyptus. - M .: Soviet Encyclopedia, 1972. - ( Great Soviet Encyclopedia : [in 30 vols.] / Ch. Ed. A. M. Prokhorov ; 1969-1978, vol. 8).
↑ Sivukhin D.V. General course of physics. - M .: Nauka , 1977 .-- T. III. Electricity. - S. 186. - 688 p.
↑ Sivukhin D.V. General course of physics. - M .: Nauka , 1977 .-- T. III. Electricity. - S. 197-198. - 688 p.

[1] Joule - Lenza law // Debtor - Eucalyptus. - M .: Soviet Encyclopedia, 1972. - ( Great Soviet Encyclopedia : [in 30 vols.] / Ch. Ed. A. M. Prokhorov ; 1969-1978, vol. 8).

[2] Sivukhin D.V. General course of physics. - M .: Nauka , 1977 .-- T. III. Electricity. - S. 186. - 688 p.

[3] Sivukhin D.V. General course of physics. - M .: Nauka , 1977 .-- T. III. Electricity. - S. 197-198. - 688 p.