Introduction:

Power is the rate of doing work and might be deduced in such units as joules per second and kilowatts. The power of a waterfall based on the height of the fall and on the number of kilogram-weights of water transferred per unit time. Likewise, in electric circuits the power expended in heating a resistor, charging a storage battery or turning a motor based on the difference of potential between the terminals of the device and the electric current via it.

The electric power is the rate of expending energy or doing work in an electrical system. For a direct-current, it is provided by the product of the current passing via a system and the potential difference across it. Electric currents can do work in numerous ways, as charging storage batteries, running electric motors and producing heat.

Current Power:

Let us assume that a current 'I' directed from an Equipotential at potential V_a to an Equipotential at potential V_b (In a time interval 'dt', a charge 'dQ' passes the V_a Equipotential while the similar amount of charge passes the V_b Equipotential. The charge passing via the area between such Equipotential thus undergoes a change in potential energy represented by:

dW = V_a dQ - V_b dQ = V_ab Idt

This is the work-done through the electric force on the charge between V_a and V_b Equipotential. The power supplied by the electric field to the charge moving between V_a and V_b Equipotential is:

P = dW/dt = VabI

Power in Electric Circuits:

The interpretation of the power equation:

P = V_abI

is of specific interest for the case whenever represents a part of an electric circuit. The electric potential energy of the circulating charges modifies at the rate V_abI as they drift via this part of the circuit having a kinetic energy which is negligibly small. In this conditions, P = V_abI is a general expression for the power input or output of this part of the circuit.

1) If V_a > V_b, the circulating charges give up energy and there is a power input.  

P_in = V_abI

2) If V_a < V_b, then the circulating charges gain energy and there is a power output.

P_out = V_abI

Power Dissipation in a Resistor, Joule's Law:

When the part of the circuit is a pure resistance 'R', the potential drop V_ab = V_a - V_b is always positive (that is, an IR drop in the direction of the current), therefore there is a power input to the resistor, P = V_abI.

Mobile charged particles are accelerated through the electric field in the conductor; however the kinetic energy gained through the charge carriers is transferred by collisions to the atoms of the conductor. The total result is that electric potential energy of the mobile charged particles is transformed into internal energy (or thermal energy) of the conductor. As the internal energy of the conductor rises, its temperature raises till there is an outward flow of heat at similar rate as the energy input. In this procedure, termed as Joule heating is the input power dissipated in the conductor.

Different expressions for the power dissipated are as follows:

P = VI = I²R = V²/R

For an ohmic resistor, the above equation is termed as Joule's law.

Electric Power and Energy:

Electrical energy 'U' is the electric power, 'P' times the time, 't':

U = Pt

The kilowatt-hour is the unit of energy. We can pay money for electrical energy at a certain price per kilowatt-hour.

Electric Power and Electromotive Force:

Let us compute the electrical power needed to charge a storage battery. You would remind that the terminal voltage of a storage battery throughout the charging procedure is bigger than its electromotive force 'E' by the amount of the internal voltage drop in the battery. Therefore when the battery charger sets up a terminal voltage 'V' in sending a charging current 'I' via the internal resistance 'r' then we have:

V = E + Ir

The power delivered to the battery is 'V' times 'I':

P = EI + I²r

Incandescent Lamp and Heating Elements:

The first incandescent lamps, developed more than a century ago, were platinum wires heated red hot via currents from the voltaic cells. The lamps had little practical utilization, both because of their small luminous efficiencies and as the batteries were costly and inconvenient. The development of the generator, based on the scientific discovery of electromagnetic induction by Michael Faraday in the year 1831, provided a reasonable source of electrical energy and led to a search for the filament materials which could be operated at a higher temperature than platinum.

The expression P = V²/R represents that for a fixed supply potential difference of V₁ the rate of heat production through a resistor rises as R reduces. Now, R = σI/A, thus P = V₂A/ρI and thus where a high rate of heat production at constant potential difference is needed, as in an electric fire on the mains, the heating element must encompass a big cross-section area 'A', a small resistivity 'P' and a short length 'l'. It should as well be capable to withstand high temperatures without oxidizing in air (and becoming brittle). Nichrome is the material that best satisfied all such needs.

The Electric lamp filaments have to operate at even higher temperatures whenever they are to emit light. In this situation, tungsten, which consists of a very high melting point (3400^oC), is used either in the vacuum or more frequently in an inert gas (nitrogen or argon). The gas decreases evaporation of the tungsten and prevents the vapor condensing on the inside of the bulb and blackening it. In modern projector lamps, there is a little iodine that makes tungsten iodide with the tungsten vapor and remains as vapor if the lamp is working, thus preventing the blackening.

Fuses:

In buildings, electrical devices are joined in parallel across the supply lines. The resistance of high-power devices is smaller than that of the low power ones. The resistance of a 30W, 220V lamp is two times that of a 60W 220 V lamp.

If electrical supply wires are accidentally short-circuited by being brought into contact with one other, the resistance of the circuit so formed might be just a few hundredths of an ohm. The current becomes extremely large and heat the wire to dangerously high temperatures. To avoid this danger, fuses are joined in series having the supply lines. A fuse is a short length of wire, often tinned copper, chosen to melt if the current via it surpasses a certain value. It thus protects a circuit from excessive currents.

If the current acquires a prescribed value, for instance 15A, the metal melts and the circuit is opened.

The fuse should be substituted after the short circuit has been repaired. The fuse wire is mounted in a receptacle that prevents the melted metal from setting fire to the surroundings.

Circuit breakers-electromagnetic gadgets which open the circuit whenever the current go beyond a preset value and that can be reset whenever the overload is removed are being increasingly utilized in place of fuses in buildings.

Note:

1) The temperature arrived by a given wire based only on the current via it and is independent of its length (given it is not so short for heat loss from the ends where it is supported, to matter).

2) The current needed to reach the melting point of the wire rises as the radius of the wire rises.

It obeys that fuses that melt at gradually higher temperatures can therefore be made from the similar material by employing wires of increasing radius.

Electrical Equivalent of Heat:

There is one instant and significant application of the results which we know, an application which serves up as a check on the accuracy of what we have done and as the other confirmation of the law of conservation of energy.

The work 'W' done (or energy expended) per unit charge in moving a charge 'q' from one point to the other is the potential difference, or voltage 'V'.

V = W/q

W = qV

W = qV = IVt

It represents the work done or the energy expended in the circuit in time 't'. This is frequently deduced as work or energy per time, or power 'P':

P = W/t = IV

The expended energy can be written in terms of the resistance 'R' of the circuit or a specific circuit element through Ohm's law, V = IR. By employing this relationship, the above equation has different forms:

W = IVt = I²Rt = V²t/R

The electrical energy used is manifested as heat energy, and is generally termed as joule heat or I²R losses, I²R equation represents how the joule heat differs with resistance: being the power or energy expended per time.

1) For a constant current 'I', the joule heat is directly proportional to the resistance, I²R.

2) For a constant voltage, 'V', the joule heat is inversely proportional to the resistance, V²/R.

The energy expended in an electrical circuit as represented by the above equation is in the units of joules. The relationship (that is, conversion factor) between joules and heat units in calories was established through James Joule from mechanical considerations - the mechanical equivalent of heat. We might remind that in his mechanical experiment, Joule had a descending weight turn a paddle wheel in a liquid. He then interrelate the mechanical (gravitational) potential energy lost by the descending weight to the heat produced in the liquid. The result was 1 cal = 4.18 J. An alike electrical experiment might be completed to find out the 'electrical equivalent of heat'. By the conservation of energy, the heat equivalents of mechanical and electrical energy are similar (that is, 1 cal = 4.18 J).

Experimentally, the quantity of electrical joule heat produced in a circuit element of resistance 'R' is measured through Calorimetry process. When a current is passed via a resistance (that is, immersion heater) in a calorimeter having water in an arrangement as described in the above figure, through the conservation of energy the electrical energy expended in the resistance is equivalent to the heat energy (joule heat) Q gained through the system:

Electrical energy = Heat gained

W = Q

IVt = (m_wc_w + m_calc_cal) (T_f-T_i)

Here the m's and c's are the masses and specific heats of the water and calorimeter cup, as pointed by the subscripts. T_f and T_i are the final and initial temperatures of the system, correspondingly.

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