# Thermal and Chemical effect of Currents

#### Heating effect of current

When a change dq passes across a potential difference V, the work done dW is given by,

dW = Vdq

This work represents the loss of potential energy of charges. The flow of charge dq in time dt is equivalent to current i:

i.e. $I = \dfrac{dq}{dt} \, \, or \, \, dq = idt$

Therefore, work done dW = V I dt

If constants current I passes for time t under a potential difference V, then

Net work done W= vit  ………………………Equation 1

According to Ohm’s law V = Ri

Therefore,

Work done = $i^2 R t = \dfrac{V^2}{R} t \cdots \text{Equation 2}$

This work is converted into energy of random thermal motion molecules of the conductor. That is the electric current through a conductor produces thermal energy in the conductor and the conductor gets heated. This phenomenon is called Joule’s heating effect of current and the heat produced is called joule’s heat.

If V is in volt, I in amp, R in ohm, then joule’s heat is equivalent to: $W = V I t = i^2 R t = \dfrac{V^2}{R} t \, \, \text{joule}$

Or Heat Produced, $Q = \dfrac{w}{j} = \dfrac{V I t}{j} = \dfrac{i^2 R t}{j} = \dfrac{ ( \dfrac{v^2}{R} ) t}{J} \text{kilocal}$

Where $J = 4.2 \times 10^3$ Joule/Kilocalorie is called the mechanical equivalent of heat.

#### Electric Power

The rate of work done is called the power and is dissipated in the form of heat.

Power dissipated, $P = \dfrac{dW}{dt} = V I = i^2 R = \dfrac{V^2}{R}$

The unit of power is watt.

#### Electric energy

The usual unit of energy is joule but for convenience a large unit kilowatt hour (KWH) is used.

In houses the electric appliances are connected in parallel and the electric energy consumed is measured in kilowatt hour.

No. of units = $\dfrac{Watt \times hours}{1000}$

#### Specification of a bulb or other electric appliances

If a bulb is specified a voltage V and power P, then resistance R and Maximum allowed current may be determined.

Power (P) = Voltage (V) X Current (i)

Therefore, maximum allowed current is bulb, $I = \dfrac{p}{V}$

Therefore, Resistance of its filament, $R = \dfrac{V}{i} = \dfrac{V}{\dfrac{p}{V}} = \dfrac{v^2}{p}$

The bulbs and other electric appliances are manufactured for parallel combination. If they are connected in series, the effect is reserved. For example if two bulbs of 25 W, 100 W are given, then in parallel 100 W bulb glows more brightly but in series the 25 W bulb would blow more brightly.

#### Characteristics of Fuse

Fuse is used for the safety of electrical appliances. It must have high resistivity and low melting point. So, it is made of tin-lead alloy.

Let R be the resistance, $\delta$ resistivity, l length, a cross sectional area and I amp its current carrying capacity.

When the fuse is safe, then for its steady state temperature heat produced per second must be equal to heat radiated by it per second.

#### Heat produced in fuse wire per second $H = \dfrac{Q}{t} = i^2 R = i^2 ( \dfrac{\delta l}{A} ) = \dfrac{i^2 \delta l}{\pi r^2} \cdots \text{equation 1}$

If e is the emissivity of the fuse material of radius r and T is the temperature, the according to Newton’s law of cooling, then energy radiated per second = $\dfrac{e 2 \pi r}{T} \cdots \text{equation 2}$

Equation 1 and 2, $e ( 2 \pi r l) T = i^2 ( \dfrac{\delta l}{\pi r^2} )$

i.e. $T = \dfrac{i^2 \delta}{2 \pi ^2 e r^3}$

Obviously the steady state temperature of a fuse is independent of length. Hence length is immaterial for an electric fuse.

Obviously for a given material of fuse wire: $i^2 \propto r^3$

### Chemical Effect of Current:

Electrolysis: The process by which a liquid is decomposed into ions is called electrolysis. The liquid which conducts electricity and undergoes decomposition is called the electrolyte. The two plates dipped in liquid and connected to battery are called electrodes. The electrode connected to positive terminal of battery is called anode while that connected to negative terminal is called cathode. The vessel containing the electrolyte and electrode is called the voltammeter.

When electrodes are connected to battery, the positive and negative ions move towards cathode and anode respectively. Therefore the positive ions are called cations while negative are called anions.

(i) The mass of ions liberated / deposited on each electrode is proportional to total charge passé through the electrolyte:

i.e. $m \propto q \\[3mm] m \propto it$

Since charge is equal to the product of current I and time t (ie. q=it) $m = Z I t \cdots \text{equation 1}$

Where Z is constant of proportionality and is called the electro-chemical equivalent.

The mass of ions deposited on each electrode is proportional to chemical equivalent of ions.

i.e. $m \propto W$

Thus if $m_1$ and $m_2$ are the masses of substances of chemical equivalent weights $W_2$ and $W_2$ respectively, then: $\dfrac{m_1}{m_2} = \dfrac{W_1}{W_2} \cdots \text{Equation 2}$

Chemical equivalent of a substance, $W = \dfrac{Atomic \, \, weight}{Valency}$

#### Electrochemical Equivalent

From Faraday’s law m=Zq, Z being electrochemical equivalent.

i.e. $Z = \dfrac{m}{q}$

If q=1 coul, Z = m

Thus the electrochemical equivalent of a substance is numerically equal to the mass of its ions liberated by the passage of 1 coulomb during electrolysis. Its units is g/coul or kg/coul.

From equation 1 and equation 2: $\dfrac{m_1}{m_2} = \dfrac{Z_1 it}{Z_2 it} = \dfrac{W_1}{W_2} = ie . \dfrac{Z_1}{Z_2} = \dfrac{W_1}{W_2}$

i.e. $\dfrac{W_1}{Z_1} = \dfrac{W_2}{Z_2} = F$

This shows that the ratio $\dfrac{W}{Z}$ is samall for all substance and is called the Faraday’s constant.

i.e. $F = \dfrac{W}{Z} \cdots \text{Equation 3}$

The value of faraday constant is 96500 coul/gram-equivalent or 96500,000 coul/kg equivalent.

### Thermoelectricity

Seebeck Effect: In 1882 Seebeck stated that “When the junctions of two dissimilar metals of two different metals are maintained at different temperatures, an EMF is induced. This effect is called Seebeck effect” and the EMF produced is called the Thermo EMF or Seebeck EMF.

Thermocouple: The device formed by connecting the ends of two different metals together to form the closed circuit is called the thermo-couple. Thermocouple

The magnitude of emf induced in the thermal couple depends on the nature of metals used. Seebeck arrangement the metals in the form of a series, called the Seebeck series. Some metalsof the series are : Bi, Ni, Co, Pt, Cu, Mn, Hg, Pb, Sn, Au, Ag, Zn, Cd, Fe, As, Sb etc

The farther are the metals in the series, greater is the emf induced. This series shows that for the same temperature difference between the junctions of the thermal couple, the emf induced in Bi-Sb couple is the maximum. The direction of current in a thermo-couple is from earlier to latter element across the hot junction. For example in Bi-Sb couple, the direction of current to form Bi to Sb across the hoy junction and in Cu-Fe couple, it is from Cu to Fe through the hot junction.

#### Dependence of Seebeck EMF with temperature

If we take a thermal couple, keep its cold junction at 0 degree Celsius and hot junction at variable temperature $t^0 C$ and plot a graph between EMF induced versus temperature of hot junction; we obtain a curve as shown in figure. The curve is parabolic in nature. Obviously with rise of temperature, the emf first increases becomes maximum and then begins to decrease becomes zero and then changes the direction. Analytically the emf induced is given by: $E = at + bt^2 \cdots \text{equation 1}$

#### Neutral temperature

The temperature at which emf becomes maximum is called neutral temperature. It is denoted bt $t_n$.

At neutral temperature, $\dfrac{dE}{dt} = 0 \, \, \text{or} \, \, a + 2 b t = 0$

Therefore,

Neutral temperature, $t_n = - ( \dfrac{a}{2b} ) ^0 C$

#### Inversion temperature

The temperature at which emf becomes zero and changes the direction is called the inversion temperature. It is denoted by $t_i$ . At inversion temperature E =0

i.e. $at + bt^2 =0$ $\text{or} \, \, t_i = - ( \dfrac{a}{b} )$

Obviously $t_i = 2t _n$ when junction is at o degree Celsius when cold junction is at $t_0 \, \, ^0 C$ , then $t_n - t_0 = t_i - t_n \\[3mm] = t_i = 2 t_n - t_0$

### The peltier Effect

Peltier effect is the inverse of Seebeck effect. It states that when a current is allowed to flow in a thermo-couple, one junction is heated and the other is cooled i.e. heat is absorbed at one junction and liberated at the other.

Remarks:

(i) Peltier effect takes place at junction only, while Joule heating takes place along the entire length of the conductors.

(ii) Peltier effect is reversible, while Joule heating is irreversible.

(iii) The heat produced or evolved at the junction is directly proportional to the current.

Peltier coefficient $\pi$: It is defined as amount of heat energy absorbed or evolved when a current of 1 ampere flows for one second. Peltier Coefficient

Thomson Effect: When a temperature difference is maintained between different parts of the same metals and a current is passed through it and then heat us either absorbed or evolved. This effect is called Thomson effect. Thomson effect is zero for lead, positive for metals below lead and negative for metals above lead in Seebeck series.

Thomson coefficient: It is defined as the amount of heat energy absorbed or evolved between two points of a conductor which differ in temperature by $1^0 C$ when a current of 1 Ampere flows for 1 second. It is denoted by $\delta$ and is also known as the specific heat of electricity.

Thermoelectric power: it is defined as the rate of change of thermo-emf with temperature:

i.e. $\dfrac{dE}{dT}$

Thermopile: It is a large number of thermocouple connected in series and is used to detect the feeble thermal radiations.

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