Physics Current Electricity

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Current Electricity

Combination of Resisters :

`text(Series combination :)`

Consider the series combination of two resistors with resistances `R_1` and `R_2` respectively as shown in the diagram.

It is obvious that they will carry the same current when connected to a battery. By equivalent of `R_1` & `R_2` between `A` and `B` in the above network, we mean a single resistor which will carry the same current for an identical potential difference across ends `A` and `B`. If `V` & `l` be the corresponding potential difference and current then for the series combination shown above,

`V=IR_1+IR_2`

If `R_(eq)` be the equivalent resistance then,

`V=I R_(eq)`

Using these equations, we get

`R_(eq)+R_1+R_2`

In general, for a series combination of n resistors the equivalent resistance will be given as

`R_(eq) = R_1 + R_2 + . . . . . + R_n`

So in series combination equivalent resistance is greater than largest individual resistance.
To get maximum resistance , resistances must be connected in series.

`text( Parallel combination :)`

For this network shown here, if `R_(eq)` be the equivalent resistance,

`V_A-V_B=I R_(eq)`

But, `I=I_1+I_2`

And `I_1=V/R_1; I_2=V/R_2`

Hence, `V/R_(eq)=V/R_1+V/R_2`

or `1/R_(eq)=1/R_1+1/R_2`

For `n` resistors in parallel, lhe equivalent resistance will be given as

`1/R_(eq)=sum_(i=1)^n 1/R`
In parallel combination equivalent resistance is lesser than smallest individual resistance.

To get minimum resistance, resistances must be connected parallel.

`text(Series combination :)`

Consider the series combination of two resistors with resistances `R_1` and `R_2` respectively as shown in the diagram.

It is obvious that they will carry the same current when connected to a battery. By equivalent of `R_1` & `R_2` between `A` and `B` in the above network, we mean a single resistor which will carry the same current for an identical potential difference across ends `A` and `B`. If `V` & `l` be the corresponding potential difference and current then for the series combination shown above,

`V=IR_1+IR_2`

If `R_(eq)` be the equivalent resistance then,

`V=I R_(eq)`

Using these equations, we get

`R_(eq)+R_1+R_2`

In general, for a series combination of n resistors the equivalent resistance will be given as

`R_(eq) = R_1 + R_2 + . . . . . + R_n`

So in series combination equivalent resistance is greater than largest individual resistance.
To get maximum resistance , resistances must be connected in series.

`text( Parallel combination :)`

For this network shown here, if `R_(eq)` be the equivalent resistance,

`V_A-V_B=I R_(eq)`

But, `I=I_1+I_2`

And `I_1=V/R_1; I_2=V/R_2`

Hence, `V/R_(eq)=V/R_1+V/R_2`

or `1/R_(eq)=1/R_1+1/R_2`

For `n` resistors in parallel, lhe equivalent resistance will be given as

`1/R_(eq)=sum_(i=1)^n 1/R`
In parallel combination equivalent resistance is lesser than smallest individual resistance.

To get minimum resistance, resistances must be connected parallel.

Methods of Determining Equivalent Resistance

(1) Method of successive reduction :
It is the most common technique to determine the equivalent resistance. So far, we have been using this method to find out the equivalent resistances. This method is applicable only when we are able to identify resistances in series or in parallel. The method is
based on the simplification of the circuit by successive reduction of the series and parallel combinations. For example to calculate the equivalent resistance between the point `A` and `B`, the network shown below successively reduced.

(2) Method of equipotential points :
This method is based on identifying the points of same potential and joining them. The basic rule to identify the points of same potential is the symmetry of the network.

(i) In a given network there may be two axes of symmetry.
(a)Perpendicular axis of symmetry, that is perpendicular to the direction of flow of current. For example in the network shown below the axis `A A ` is the parallel axis of symmetry, and the axis `B B ` is the perpendicular axis of symmetry.
(b) Parallel axis of symmetry, that is, along the direction of current flow.

(ii) Points lying on the perpendicular axis of symmetry may have same potential. In the given network, point `2, 0` and `4` are at the same potential.

(iii) Points lying on the parallel axis of symmetry can never have same potential.

(iv) The network can be folded about the parallel axis of symmetry, and the overlapping nodes have same potential. Thus as shown in figure, the following points have same potential.
(a) `5` and `6`
(b) `2, 0` and `4`
(c) `7` and `8`

(1) Method of successive reduction :
It is the most common technique to determine the equivalent resistance. So far, we have been using this method to find out the equivalent resistances. This method is applicable only when we are able to identify resistances in series or in parallel. The method is
based on the simplification of the circuit by successive reduction of the series and parallel combinations. For example to calculate the equivalent resistance between the point `A` and `B`, the network shown below successively reduced.

(2) Method of equipotential points :
This method is based on identifying the points of same potential and joining them. The basic rule to identify the points of same potential is the symmetry of the network.

(i) In a given network there may be two axes of symmetry.
(a)Perpendicular axis of symmetry, that is perpendicular to the direction of flow of current. For example in the network shown below the axis `A A ` is the parallel axis of symmetry, and the axis `B B ` is the perpendicular axis of symmetry.
(b) Parallel axis of symmetry, that is, along the direction of current flow.

(ii) Points lying on the perpendicular axis of symmetry may have same potential. In the given network, point `2, 0` and `4` are at the same potential.

(iii) Points lying on the parallel axis of symmetry can never have same potential.

(iv) The network can be folded about the parallel axis of symmetry, and the overlapping nodes have same potential. Thus as shown in figure, the following points have same potential.
(a) `5` and `6`
(b) `2, 0` and `4`
(c) `7` and `8`

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