Question

Derive an expression for equivalent resistance for parallel combination of resistance

Answer 1

The figure shows a circuit with resistors $\sf{R_1,\ R_2}$ and $\sf{R_3}$ connected in parallel combination.

Let the amount of current flowing through the resistor,

• $\sf{R_1}$ be $\sf{I_1}$

• $\sf{R_2}$ be $\sf{I_2}$

• $\sf{R_3}$ be $\sf{I_3}$

and the net current flowing through the circuit be I.

Since the current flows from A to B but horizontally, the potential difference in each resistor vary horizontally due to change in position, but not vertically. Therefore the potential difference in each resistor $\sf{R_1,\ R_2}$ and $\sf{R_3}$ is the same, as they're vertically differ in their positions only but not horizontally. Let each be V.

Hence by Ohm's Law,

• $\sf{I=\dfrac{V}{R}}$

• $\sf{I_1=\dfrac{V}{R_1}}$

• $\sf{I_2=\dfrac{V}{R_2}}$

• $\sf{I_3=\dfrac{V}{R_3}}$

Then the net current flowing through the circuit is given by,

$\longrightarrow\sf{I=I_1+I_2+I_3}$

$\longrightarrow\sf{\dfrac{V}{R}=\dfrac{V}{R_1}+\dfrac{V}{R_2}+\dfrac{V}{R_3}}$

$\longrightarrow\sf{\dfrac{1}{R}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}}$

$\longrightarrow\sf{\underline{\underline{R=\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}\right)^{-1}}}}$

$\longrightarrow\sf{\underline{\underline{R=\dfrac{R_1R_2R_3}{R_1R_2+R_2R_3+R_3R_1}}}}$

This is the equivalent resistance of three resistors in parallel combination.

Equivalent resistance of 'n' resistors $\sf{R_1,\ R_2,\ R_3,\dots\ R_n}$ in parallel combination is,

$\longrightarrow\sf{\underline{\underline{R=\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+\dots+\dfrac{1}{R_n}\right)^{-1}}}}$

Answer 2

Explanation:

Answer. (i) Total potential difference across a combination of resistors in series is equal to the sum of potential difference across the individual resistors. Hence , a fuse of rating 5A can be used because it can bear the current

Answer. (i) Total potential difference across a combination of resistors in series is equal to the sum of potential difference across the individual resistors. Hence , a fuse of rating 5A can be used because it can bear the current