Answer
Verified
427.8k+ views
Hint: Resistors are said to be in a parallel state when both the terminals of the resistors are connected to each terminal of the other resistors. The resistors in parallel have a common voltage across all the resistors. Here, we will use four resistors connected in parallel to find the law of combination of resistances.
Complete answer:
The figure below shows the circuit in which the resistors are placed parallel to each other.
Let ${R_1}$ , ${R_2}$ , ${R_3}$ and ${R_4}$ are resistors that are connected parallel to each other. Here, the voltage drop across each resistor will be the same but the electric current in the circuit will divide itself to travel through all the different branches. Now. to derive the equation of resistance in parallel, we will use Ohm’s law which is given by
$V = IR$
$I = \dfrac{V}{R}$
Here, $I$ is the current in the circuit, $V$ is the voltage, and $R$ is the resistance in the circuit.
Now, according to Kirchhoff’s law, we get
$\sum {{I_{in}}\, = \,\sum {{I_{out}}} } $
$ \Rightarrow \,I = {I_1} + {I_2}$
$I = \dfrac{{{V_1}}}{{{R_1}}} + \dfrac{{{V_2}}}{{{R_2}}}$
Since the voltage drop across the resistors is the same. Therefore,
$I = \dfrac{V}{{{R_1}}} + \dfrac{V}{{{R_2}}}$
$ \Rightarrow \,I = V\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
$ \Rightarrow \,\dfrac{I}{V} = \left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
$ \Rightarrow \,{R_p} = {\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)^{ - 1}}$
Therefore, the resistance in the parallel series are
$\therefore{R_p} = {\left( {\dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + ...... + \dfrac{1}{{{R_{n - 1}}}} + \dfrac{1}{{{R_n}}}} \right)^{ - 1}}$
This means that the combined resistance in the parallel circuit will be equal to the sum of the reciprocal of all the individual resistances.Therefore, the law of combination of resistances in parallel states that the reciprocal of the combined resistance of all the resistors connected in parallel is equal to the sum of the reciprocal of all the individual resistance.
Note:In a parallel resistor circuit, the voltage across all the resistors will be the same. Therefore, the voltage in the resistor ${R_1}$ will be equal to the voltage in the resistor ${R_2}$ and is also equal to the voltage in the resistor ${R_3}$ . That is why we have used the same voltage for all the resistances.
Complete answer:
The figure below shows the circuit in which the resistors are placed parallel to each other.
Let ${R_1}$ , ${R_2}$ , ${R_3}$ and ${R_4}$ are resistors that are connected parallel to each other. Here, the voltage drop across each resistor will be the same but the electric current in the circuit will divide itself to travel through all the different branches. Now. to derive the equation of resistance in parallel, we will use Ohm’s law which is given by
$V = IR$
$I = \dfrac{V}{R}$
Here, $I$ is the current in the circuit, $V$ is the voltage, and $R$ is the resistance in the circuit.
Now, according to Kirchhoff’s law, we get
$\sum {{I_{in}}\, = \,\sum {{I_{out}}} } $
$ \Rightarrow \,I = {I_1} + {I_2}$
$I = \dfrac{{{V_1}}}{{{R_1}}} + \dfrac{{{V_2}}}{{{R_2}}}$
Since the voltage drop across the resistors is the same. Therefore,
$I = \dfrac{V}{{{R_1}}} + \dfrac{V}{{{R_2}}}$
$ \Rightarrow \,I = V\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
$ \Rightarrow \,\dfrac{I}{V} = \left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
$ \Rightarrow \,{R_p} = {\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)^{ - 1}}$
Therefore, the resistance in the parallel series are
$\therefore{R_p} = {\left( {\dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + ...... + \dfrac{1}{{{R_{n - 1}}}} + \dfrac{1}{{{R_n}}}} \right)^{ - 1}}$
This means that the combined resistance in the parallel circuit will be equal to the sum of the reciprocal of all the individual resistances.Therefore, the law of combination of resistances in parallel states that the reciprocal of the combined resistance of all the resistors connected in parallel is equal to the sum of the reciprocal of all the individual resistance.
Note:In a parallel resistor circuit, the voltage across all the resistors will be the same. Therefore, the voltage in the resistor ${R_1}$ will be equal to the voltage in the resistor ${R_2}$ and is also equal to the voltage in the resistor ${R_3}$ . That is why we have used the same voltage for all the resistances.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE