Question

The practical unit of resistance is \Omega ( $ \Omega $ ). What is $ 1\Omega $ equal to?
(A) $ {10^{18}}emu $
(B) $ {10^9}emu $
(C) $ {10^{15}}emu $
(D) none of these

Answer 1

Hint
Use the conversion of electric potential into a CGS emu system of units as $ 1V = {10^8}emu $ of potential and for current use the conversion as $ 1A = 0.1emu $ of current. And, use the definition of $ 1\Omega $ in terms of volts and amperes to convert it to emu-cgs units.

Complete step by step answer
We know that, converting electric potential into CGS emu system of units we obtain,
$ \Rightarrow 1V = {10^8}emu $ of voltage ... (1)
Converting electric current into emu-cgs system of units, we obtain,
$ \Rightarrow 1A = \dfrac{1}{{10}}emu $ of current … (2)
Also, by definition of $1 \Omega$, we know that $1 \Omega$ is equal to the resistance of a conductor through which a current of one ampere flows when a potential difference of one volt is applied to it.
Therefore, by \Omega’s law, we can express it mathematically as,
$ \Rightarrow 1\Omega = \dfrac{{1V}}{{1A}} $
Substituting the value of $1V$ from Equation (1) and the value of $1A$ from Equation (2) in the above equation, we get,
$ \Rightarrow 1\Omega = \dfrac{{{{10}^8}emu(V)}}{{\dfrac{1}{{10}}emu(I)}} $ … (3)
In equation (3), $ emu(V) $ is the emu of voltage and $ emu(I) $ is the emu of current. And we have the relationship between them as follows
$ \Rightarrow \dfrac{{emu(V)}}{{emu(I)}} = emu(R) $ where, $ emu(R) $ is the emu of Resistance, which we will refer to as $ emu $ .
Hence, Equation 3 can be written as:
$ \Rightarrow 1\Omega = \dfrac{{{{10}^8}}}{{\dfrac{1}{{10}}}}emu $
Simplifying the above equation by rearranging and adding the powers of 10, we get,
$ \Rightarrow 1\Omega = {10^8}.10 = {10^{8 + 1}}emu $
Hence, we obtain the relationship between $1\Omega$ and $1emu$ of resistance as,
$ \Rightarrow 1\Omega = {10^9}emu $ .
Option (B) is correct.

Note
emu or electro-magnetic unit is one of the various extensions to the cgs system of units to electromagnetism. The others include – Electrostatic CGS unit (ESU), Gaussian CGS units and Lorentz-Heaviside CGS units. Students should not confuse different emu-cgs units used in equation 3 as the same unit. Emu of Potential is different from emu of current (also called abampere) and that is different from emu of resistance. Also, emu of charge is known as abcoulomb.