Series and Parallel Circuits

1.      What is an electrical circuit?

a.      An electric circuit is the path or network of paths formed by electrical components through which electric current flows when the circuit is closed.

b.      Electrical components that form a circuit with an electric source include:

                          i.      Resistor

                        ii.      Lamp

                      iii.      Switch

                      iv.      Electrical appliances

                        v.      Connecting wires

c.       Internal and external circuits:

                          i.      Internal circuit refers to the path taken by the current within the source (cell, battery, etc.).

                        ii.      External circuit refers the path taken by the current outside the source.

d.      Series and parallel circuits are 2 basic ways of connecting electrical components to form circuits for the current to flow.

2.      A series circuit refers to the single path formed by electrical components which are connected end to end consecutively to an electric source.

3.      In a series circuit, experimental evidence (at pg. 368) shows that:

a.       Potential difference (PD a.k.a. V):

                                       i.      PD applied across the external circuit = the SUM of individual PDs across each external component: V = V1 + V2 + V3 +…;

                                     ii.      In other words, the external components share the applied voltage across the external circuit;

                                    iii.      The electromotive force V(e.m.f.) or E of the cell (or battery) = the Sum of individual PDs across all components (including across the cell): V(e.m.f.) = V(drop) + V1 + V2 + …

                                   iv.      In other words, all the components in the circuit (incl. the cell) share the V(e.m.f.) of the cell.

b.      Current (I)

      SAME current flows through all the components: I = I₁ = I₂ = I₃ =…;

(Yr 2006 SPM P1 Q42 at pg. 58)

(Yr 2009 SPM P1 Q38 at pg. 194)

(Yr 2010 SPM P1 Q39 at pg. 240)

c.       Resistance (R):

                                       i.      Effective Resistance = the SUM of individual resistance of each component: R = R₁ + R₂ + R_3 ... (Also confirmed by: IR = IR₁ + IR₂ + IR₃ + …).

                                     ii.      In other words:

1.      the effective resistance R in a series circuit is larger than each of the individual resistor;

2.      the combination of resistors in series effectively forms a longer resistor with higher resistance

                  (Yr 2006 SPM P1 Q39 at pg. 57)

4.      A parallel circuit refers to the network of separate and parallel paths formed by electrical components which are connected side by side and their corresponding ends are joined together to an electric source, a cell or a battery.

5.      In a parallel circuit, experimental evidence (at pg. 368) shows that:

1. Potential difference (PD a.k.a. V):

                                                               i.      SAME PD (potential difference) across separate pathways: V = V₁ = V₂ = …;

                                                             ii.      In other words, voltage across each resistor in parallel is the same.

   (Yr 2005 SPM P1 Q43 at pg. 13)

(Yr 2007 SPM P1 Q36 at pg. 99)

2. Current (I):

                                                               i.      The SUM of currents in separate pathways = Total current leaving or returning to cell: I = I₁ + I₂ + I₃ + …;

                                                             ii.      In other words, resistors in parallel share the main current.

                                                            iii.      As current in each pathway is given by V/R, and V is the same for all pathways, therefore, main current = sum of individual currents also means:

               V/R = V/R₁ + V/R₂ + V/R₃ + … (This leads us to the equation for effective resistance in a parallel circuit: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + …)

3. Resistance (R):

                                                               i.      Effective resistance of resistors in parallel is given by: The reciprocal of the effective resistance = sum of the reciprocals of individual resistance in each pathway:

               1/R = 1/R₁ + 1/R₂ + 1/R₃ + …

               (From: V/R = V/R₁ + V/R₂ + V/R₃ + …, please see 5b(iii).)

            (Yr 2008 SPM P1 Q36 at pg. 147)

                                                             ii.      In other words:

1.      the effective resistance R in a parallel circuit is smaller than each of individual resistor;

2.      the combination of resistors in parallel effectively forms a resistor with larger cross-sectional area and therefore lower combined resistance.

                                                            iii.      When identical resistors are in parallel:

1.      Quick Formula to Calculate Effective Resistance:

      R(effective) = R(individual)/n

      Where, n = number of identical resistors in parallel

(Yr 2006 SPM P1 Q38 at pg. 57)

(Yr 2008 SPM P1 Q37 at pg. 148)

(Yr 2011 SPM P2 Q6 at pg. 301~302)

6. Combined circuit refers to a circuit with series and parallel arrangements of components.

(Yr 2010 SPM P1 Q40 at pg. 241)

(Yr 2010 SPM P2 CQ12(b) at pg. 264~266)

Summary:

Ohm's law states that the current flowing, I through an ohmic conductor (pure metal) is directly proportional the potential difference, V across the conductor if the temperature remains constant.

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Example how to determine effective resistance of a parallel circuit:

• When resistors are connected in parallel, the effective resistance becomes smaller compare to that of connection in series.
• Thus, if terminal voltage Vt is the same, a higher main current will be shared by the individual resistors in parallel - thus each resistor will receive higher flow of current thereby producing a brighter bulb in parallel circuit
  
  

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Updated by: tutortan1@gmail.com (25/05/16)

Potential Difference, Current and Resistance

1.      Just as an object placed at different depths of water experiences different water pressures, or, an object at different distances from the centre of the earth carries different gravitational potential energies, a charged particle placed at different points of an electric field also experiences different electric forces acting on it and carries different electric potential energies.

2.      The potential difference or voltage between 2 points in an electric field is defined as the work done in moving 1 coulomb of charge from 1 point to the other. Suppose, work, W is done to move a charge, Q from 1 point to another, the potential difference, V is given by:

Potential difference = Work done / Charge;

V = W / Q

3.      The potential difference (a.k.a. voltage) between  2 points is 1 volt if 1 joule of work is required to move a charge of 1 coulomb from 1 point to the other:

1 volt = 1 joule / 1 coulomb = 1 J C^-1 = 1 V (unit symbol for volt)

4.      Just as it is the pressure difference that causes water to move from a region of higher pressure to a region of lower pressure, or an object to move from high ground to low ground due to different gravitational potential energies, a charged particle too would move from a point of higher electric potential (+ve) to another point of lower electric potential (-ve) in an electric field or circuit – due to the potential difference between the 2 points.

5.      The movement of charged particles from 1 point to another in an electric field or a circuit due to the potential difference between the 2 points produces an electric current. The direction of flow of current is by convention also from positive to negative. The rate of flow of the charge determines the size of the current. But what determines the rate of flow of the charge? Read on…

6.      Ohm’s law (by a German physics teacher, Georg Simon Ohm in 1826) states that the current flowing through an ohmic conductor is directly proportional to the potential difference across its ends provided that its temperature and other physical conditions (length, cross-sectional area and material type) remain constant:

I = Constant x V; or,

V = Another constant x I (This other constant is known as the resistance, R)

V = R x I = IR (R is the gradient of a linear V-I graph for ohmic conductor); or

R = V/I (R is the ratio of instantaneous voltage over current for all conductors); or

I = V/R (I is also the ratio of voltage over resistance)

(Yr 2010 SPM P1 Q38 at pg. 240)

7.      Thus, the V-I graphs:

a.       of all ohmic conductors are linear graphs which pass through the origin and their gradients are the resistances R of the respective ohmic conductors – the steeper the gradient, the higher the resistance;
                    

b.      of non-ohmic conductors are non-linear which pass through the origin and the ratio of V over I at any point of the V-I graph gives the value of resistance at that point – the higher the ratio, the higher the resistance.

(Yr 2006 SPM P1 Q36 at pg. 57)

8.      Resistance, R of a conductor is therefore defined as the ratio of the potential difference V across the conductor to the current I flowing through it. This applies to both ohmic and non-ohmic conductors. The unit of measurement of resistance R is therefore volt per ampere (V A^-1) or ohm (Ω).

9.      Factors that affect the resistance R of a conductor are (experiments at pgs 357 ~ 362):

c.       Its length, l – (directly proportional) the longer the length, the higher the resistance;

d.      Cross-sectional area, A – (inversely proportional) the bigger the area, the lower the resistance;

e.       The type of material – resistivity ρ depends on material; and,

f.        Its temperature :

·        Most Pure metal - (proportional): the higher the temperature, the higher the resistance.

·        Alloys (constantan, nichrome) – resistance increases slightly with temperature increases.

·        Thermistor – resistance decreases greatly with slight increase in temperature.

·        Superconductor – resistance becomes zero at critical low temperature.

(Yr 2007 SPM P1 Q35 at pg. 99)

(Yr 2009 SPM P1 Q37 at pg. 194)
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More on Thermistor (Negative Temperature Coefficient, Thermistor)

Thermistors work by translating temperature into resistance, with resistance decreasing as temperature increases (referred to as a 'negative temperature coefficient’, or NTC, thermistors).

The graph below illustrates the resistance of the thermistor as a function of the temperature:

Thermistor Resistance vs. Temperature Graph


As can be seen from the graph, the resistance of the thermistor drops very quickly in the temperature range 0°C to 40°C - it offers good sensitivity to changes in temperature in this range; however, at much higher temperatures, it will be less sensitive to temperature changes.

10.  The resistance R of a resistor of a given material and at a given temperature can be calculated using this relationship: R = ρ l/A, where R is directly proportional to length l and resistivity ρ of the resistor and is inversely proportional to cross-sectional area A of the same. Resistivity ρ of a resistor at a given temperature is a constant dependent on the material of the resistor. Thus,

                        R = ρ l/A; or,

                        ρ = RA/l

11.  Voltmeter:

g.       It measures potential difference or voltage in volts (V);

h.       It is connected in parallel across the resistor or device;

i.         It has high resistance so that the current flowing through it is negligible.

12.  Ammeter (or milliammeter):

j.        It measures current in amperes (A) (or milliamperes, mA);

k.      It is connected in series with the resistor or component;

l.         It has low resistance so that its existence has insignificant effect on the magnitude of current flowing and to be measured.

13.  Measurement of Resistance: To measure resistance, we usually take the reading of the voltmeter in volts across the resistor over the reading of the ammeter for the current flowing through the resistor. Thus, R = V/I    (Instead of using the formula: R = ρ l/A, which we can use too if resistivity ρ, length l and cross sectional area A are all readily and accurately measurable or available) 

14.  Superconductors:

m.   What are superconductors? Superconductors are materials which offer no resistance (i.e. zero resistance) to the flow of current when they are cooled to below certain temperatures known as the critical temperatures for superconductivity.

n.       Only some metals show superconductivity, for examples:

Name of Elements              Critical Temperature (K)

Zinc, Zn                                          0.88

Aluminium, Al                                 1.14

Tin, Sn                                           3.69

Mercury, Hg                                   4.15

Lead, Pb                                        7.26

Niobium, Nb                                  9.2

(Some of the best conductors of electricity at normal temperature like copper, silver, gold are not superconductors even at absolute zero temperature, 0 K although their resistance R decreases with temperature)

o.      Once current flows in superconductors, it needs no further applied voltage (electric energy per coulomb) to persist flowing – there is no loss of current.

p.      Superconductors can produce magnets with magnetic field strengths > 10 times that of the best normal electromagnets. These superconducting magnets are useful:

                                                   i.      in the development of magnetically levitated trains and vehicles of the future;

                                                 ii.      in Magnetic Resonance Imaging (MRI) scanner as diagnostic tool in medicine

                                                iii.      to produce computer chips which are faster and smaller.

q.      Superconducting wires or cables increase the efficiency of electrical power transmission as loss of energy as heat is greatly reduced.

r.        Students must be able to recognize the R (Resistance) – T (Temperature) graphs of normal conductors (pl see below for copper), NTC thermistor (pl see below), RTD (below) and those of superconductors (pg. 365).

                  
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By: tutortan1@gmail.com (edited on 24/05/16)