Question

Numerically \[1\,{\text{gauss}} = x\,{\text{tesla}}\], then \[x\] is
A. \[{10^{ - 4}}\]
B. \[{10^4}\]
C. \[{10^8}\]
D. \[{10^{ - 8}}\]

Answer 1

Hint:We are asked to find the value of \[x\]. At first know the difference between gauss and tesla, recall what these units represent. Recall which is the CGS unit and which is the SI unit. Using a suitable formula try to convert the CGS unit into SI unit and find the value of\[x\].

Complete answer:
Gauss is the CGS unit and tesla is the SI unit for measurement of magnetic field strength.
We are asked to find the value of \[x\] in the equation \[1\,{\text{gauss}} = x\,{\text{tesla}}\].
To find out their conversion factor, let us find the unit of magnetic field strength in both CGS and SI units.

Lets us take the formula of magnetic Lorentz force which is given as,
\[\overrightarrow F = q(\overrightarrow v \times \overrightarrow B ) \\
\Rightarrow F = qvB\sin \theta \]
\[ \Rightarrow B = \dfrac{F}{{qv\sin \theta }}\] (i)
where \[\overrightarrow F \] is the force on the particle, \[q\] is the charge on the particle, \[\overrightarrow v \] is the velocity , \[\overrightarrow B \] is the magnetic field and \[\theta \] is the angle between the velocity of particle and the magnetic field.
We now use dimensional analysis in equation (i),
\[\left[ B \right] = \dfrac{{\left[ F \right]}}{{\left[ q \right]\left[ v \right]}}\] (ii)
The CGS unit of magnetic field is gauss, force is dyne and velocity is centimetre per second. Using these facts in equation (ii), we get
\[1\,{\text{gauss}} = \dfrac{{1\,{\text{dyne}}}}{{\left( {1\,{\text{ampere - second}}} \right)\left( {1\,{\text{cm - second}}} \right)}}\] (iii)
Now, we convert the quantities on the R.H.S to SI units.
\[1\,{\text{dyne}} = {10^{ - 5}}{\text{N}}\], \[{\text{1 cm}} = {10^{ - 2}}\,{\text{m}}\] and the unit of charge in SI unit is \[{\text{10}} \times ({\text{CGS unit)}}\]. Using these values in equation (iii), we get
\[1\,{\text{gauss}} = \dfrac{{{{10}^{ - 5}}{\text{N}}}}{{\left( {10\,{\text{ampere - second}}} \right)\left( {{{10}^{ - 2}}\,{\text{m - second}}} \right)}}\]
\[ \Rightarrow 1\,{\text{gauss}} = \dfrac{{{{10}^{ - 4}}{\text{N}}}}{{(1\,{\text{ampere - second}})({\text{m - second}})}}\]
We have SI unit of magnetic field as tesla,
\[\therefore 1\,{\text{gauss}} = {10^{ - 4}}\,{\text{tesla}}\]
Therefore, comparing this equation with the given equation we get, \[x = {10^{ - 4}}\].

Hence, the correct answer is option A.

Note: Remember there are three fundamental units : length \[{\text{L}}\], mass \[{\text{M}}\], time \[{\text{T}}\]. There are two types of units; SI unit which is International system of Units and CGS unit which is Centimetre Gram Second unit. SI units of length, mass and time are metre, kilogram and second respectively and CGS units of length, mass and time are centimetre, gram and second respectively.