A Discourse on Simple Direct Current Circuits: Resistors

In electric circuit, there are usually active and passive components and these two can be distinguished thus. Any two-terminal element or component which can supply power continuously to an external load is said to be active. And they are also described as generators. On the other hand, a passive component refers to any component that only consumes or stores energy. Examples are ordinary connecting wires, resistors and capacitors.

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For steady currents the two types can be distinguished by measuring the potential difference between the terminals on open circuits. If the result is zero, then the correspondent is passive. The definition of resistance R in R=V/I does not apply to active components.

For example in chemical cells the potential difference V between the terminals reduces with the current I supplied. For sufficient small current this variation is often such that dV/dI remains constant, and the internal resistance of the cells is given by

        R = -dV/dI

If the cells is used under conditions for which r is independent of V or I, then a relationship between V and I may be obtained by equation

      dV/dI     =  -r
      dV        =  -rdl

So that we have V = -rI + constant.

The constant is given by the value of V when the current I is zero, that is, by the open circuit terminal potential difference, which we already know as electromotive force (EMF), E of the cell. Therefore,

      V = E - rI

In terms ofCombination of Resistors, resistors are normally interconnected in a circuit in a complicated manner; and in order to analyze the circuit, it is often necessary to determine the resistance of the simple resistor which would produce the original combination in the circuit as the original combination of resistors. This single resistance is called the equivalent resistance of the combination, Req.

Resistors in Series Combination

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Resistors are said to be connected in series if they are joined end-to-end so that the same current flows through each of them. In a series combination, V1, V2, V3,.....Vn are the potential differences accross the resistors, R1, R2, R3, ....., Rn respectively when current I flows. Thus,

 V = V1 + V2 + V3 + .... + Vn

Hence,

V = IR1 + IR2 + IR3 + .... + IRn

If R is the equivalent resistance of the combination then

V = IR

Hence,, from the above,

IR = IR1 + IR2 + IR3 + .... + IRn

Therefore,

R = R1 + R2 + R3 + .... + Rn

Thus, the equivalent resistance of a series combination is the sum of the individual resistances.

For resistors in Parallel, resistors are said to be connected in parallel if they all have common terminals.

Let I1, I2, I3..., In be the current flowing in R1,R2 ,R3...., Rn respectively when current I enters the combination.

Clearly,

I = I1+I2+I3+...+In
I = V/R1 + V/R2 + V/R3 + ... + V/Rn

If R is the equivalent resistance of the combination then we can say that
I= V/R.
Hence, from the above equations, we have

V/R = V/R1 + V/R2 + V/R3+... + V/Rn

Therefore,

1/R1 + 1/R2 + 1/R3 +...+ 1/Rn

Thus, the reciprocal of the equivalent resistance of a parralell combination is the sum of the reciprocal of the individual resistances. In terms of total current I , the branch current of a parrallel combination are given, from, by

I1 = V/R1 = IR/R1
I2 = V/R2 = I/R2
I3 = V/R2 = IR3
In = V/R = IR/RN

We then see that the branch current are inversely proportional to the branch resistances.

From the foregoing, it is known that cells can be connected in Series or in Parrallel. In the series arrangements, the positive pole of one cell is connected to the negative of next cell. But in the parallel combination, all the positive poles are joined together and so are all the negative poles.

Furthermore, experiments show that if E is total EMF supplied by the cells and E,E2, etc are the individual emfs then

E = E1 + E2 + E3 + ... + En

In general, the total e.m.f, supplied by cells connected of cells in essence is equal to the sum of their individual e.m.f.s. If the cells have internal resistance,then the combination of cells in series has the internal resistances in series. The two statements can be summarized as:

E= E1+E2+E3+....+En

and

R= r1 + r2 + r3 + .... rn

If the cells are in Parrallel,the resultant e.m.f is that of one cell only. If two accumulators and one Leclanche cell were connected in parallel,the accumulators would drive a current through the Leclanche cell as the e.m.f of the accumulators is heigher than the e.m.f of a Leclanche cell.

For parallel combinations, only cells of the same type are used, otherwise,a current may be passed through a cell in the wrong direction and causes damage to it. With cells connected in parrallel,the current divides to pass through each cell, hence each cell passes less current than a cell by itself. Parallel combinations of cells are therefore used when high current are needed in a circuit. The total internal resistance of cells is lower when they are connected in parallel.

It is worthy of note that thr Voltmeter is not accurate for absolute determination of potential difference or electrical force as they possess a resistance although the resistance is high. A perfect voltmeter should have an infinitely high resistance, but this is not possible. An absolute measurements of potential difference or electromotive force is made by a potentiometer.

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References

https://en.m.wikipedia.org/wiki/Resistor

https://www.electronics-tutorials.ws/resistor/res_3.html

https://openpress.usask.ca/physics155/chapter/6-2-resistors-in-series-and-parallel/

https://en.m.wikipedia.org/wiki/Electronic_component

https://www.electrical4u.com/active-and-passive-elements-of-electrical-circuit/