# State and prove the joule's law of heating?

Joule’s law of heating states that when a current ‘i ' passes through a conductor of resistance ‘r’ for time ‘t’ then the heat developed in the conductor is equal to the product of the square of the current, the resistance and time.

H = i 2 rt

The reason behind the heat produced is the collision of the free electrons with the lattice ions or atoms while accelerating in presence of the external electric field.

we know that

volt= work done/ total charge or v = w/q

from this we get w= vq

we know that current i = q / t

so q = it

putting this in prev equation

we get w = vit

from ohms law v = ir

putting it in prev eq we get

w = irit = i2rt joule as work is stored as heat so H = I2RT Joules

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## Formulas and proof

### Direct current

The most general and fundamental formula for Joule heating is:

$P=VI$

where

• P is the power (energy per unit time) converted from electrical energy to thermal energy,
• I is the current traveling through the resistor or other element,
• V is the voltage drop across the element.

The explanation of this formula (P=VI) is:[1]

(Energy dissipated per unit time) = (Energy dissipated per charge passing through resistor) × (Charge passing through resistor per unit time)

When Ohm's law is also applicable, the formula can be written in other equivalent forms:

$P=IV=I^2R=V^2/R$

where R is the resistance.

### Alternating current

When current varies, as it does in AC circuits,

$P(t)=I(t)V(t)$

where t is time and P is the instantaneous power being converted from electrical energy to heat. Far more often, the average power is of more interest than the instantaneous power:

$P_{avg}=I_{rms}V_{rms}=I_{rms}^2R=V_{rms}^2/R$

where "avg" denotes average (mean) over one or more cycles, and "rms" denotes root mean square.

These formulas are valid for an ideal resistor, with zero reactance. If the reactance is nonzero, the formulas are modified:

$P_{avg} = I_{rms}V_{rms}cosphi = I_{rms}^2 operatorname{Re}(Z) = V_{rms}^2 operatorname{Re}(Y^*)$

where $phi$ is the phase difference between current and voltage, $operatorname{Re}$ means real partZ is the complex impedance, and Y* is the complex conjugate of the admittance (equal to 1/Z*).

For more details in the reactive case, see AC power.

### Differential Form

In plasma physics, the Joule heating often needs to be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.

$P=mathbf{J} cdot mathbf{E}$

Here, $mathbf{J}$ is the current density, and $mathbf{E}$ is the electric field.

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oule heating, also known as ohmic heating and resistive heating, is the process by which the passage of an electric currentthrough a conductor releases heat. The amount of heat released is proportional to the square of the current such that

$Q propto I^2 cdot R$

### The most general and fundamental formula for Joule heating is:

$P=VI$

where

• P is the power (energy per unit time) converted from electrical energy to thermal energy,
• I is the current traveling through the resistor or other element,
• V is the voltage drop across the element.

The explanation of this formula (P=VI) is:[1]

(Energy dissipated per unit time) = (Energy dissipated per charge passing through resistor) × (Charge passing through resistor per unit time)

When Ohm's law is also applicable, the formula can be written in other equivalent forms:

$P=IV=I^2R=V^2/R$

where R is the resistance.

### Main article:AC power

When current varies, as it does in AC circuits,

$P(t)=I(t)V(t)$

where t is time and P is the instantaneous power being converted from electrical energy to heat. Far more often, the average power is of more interest than the instantaneous power:

$P_{avg}=I_{rms}V_{rms}=I_{rms}^2R=V_{rms}^2/R$

where "avg" denotes average (mean) over one or more cycles, and "rms" denotes root mean square.

These formulas are valid for an ideal resistor, with zero resistance. If the resistance is nonzero, the formulas are modified:

$P_{avg} = I_{rms}V_{rms}cosphi = I_{rms}^2 operatorname{Re}(Z) = V_{rms}^2 operatorname{Re}(Y^*)$

where $phi$ is the phase difference between current and voltage, $operatorname{Re}$ means real partZ is the complex impedance, and Y* is the complex conjugate of the admittance (equal to 1/Z*).

For more details in the reactive case, see AC power.

### In plasma physics, the Joule heating often needs to be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.

$P=mathbf{J} cdot mathbf{E}$

Here, $mathbf{J}$ is the current density, and $mathbf{E}$ is the electric field.

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