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Question: In the formula \(x = 3y{z^2}\), \(x\) and \(y\) have dimensions of capacitance and magnetic inductio...

In the formula x=3yz2x = 3y{z^2}, xx and yy have dimensions of capacitance and magnetic induction field strength respectively. Then the dimensions of yy in the system is
A.M−3L−2T4Q4{M^{ - 3}}{L^{ - 2}}{T^4}{Q^4}
B.M−2L−2T2Q2{M^{ - 2}}{L^{ - 2}}{T^2}{Q^2}
C.M−3L−1T4Q4{M^{ - 3}}{L^{ - 1}}{T^4}{Q^4}
D.M2L2T−3Q−1{M^2}{L^2}{T^{ - 3}}{Q^{ - 1}}

Explanation

Solution

Dimensional formula gives us the relation between fundamental units and derived units in terms of dimensions is called dimensional formula.
Capacitance is defined as the ratio of amount of electric charge to difference in electric potential stored in a conductor.
Magnetic induction is the production of electromotive force across electrical conductors in a changing magnetic field.

Complete answer:
According to the question,
x=3yz2x=3y{{z}^{2}}
Where xx and zz have dimensions of capacitance and magnetic induction field strength respectively.
C=Q2EC=\dfrac{{{Q}^{2}}}{E}
Where Q is charged and E is energy.
⇒C=Q2M1L2T−2\Rightarrow C=\dfrac{{{Q}^{2}}}{{{M}^{1}}{{L}^{2}}{{T}^{-2}}}
⇒C=M−1L−2T2Q2\Rightarrow C={{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}}
⇒x=M−1L−2T2Q2\Rightarrow x={{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}}
The formula for magnetic induction is
B=μ04π×2mr3B=\dfrac{{{\mu }_{0}}}{4\pi }\times \dfrac{2m}{{{r}^{3}}}
Where μ0{{\mu }_{0}} is magnetic susceptibility.
m is magnetic moment and r is radius.
B=[M1L1T−2A−2]×[L2A1][(L1)3]B=\dfrac{\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}}{{A}^{-2}} \right]\times \left[ {{L}^{2}}{{A}^{1}} \right]}{\left[ {{\left( {{L}^{1}} \right)}^{3}} \right]}
⇒B=[M1T−2A−1]\Rightarrow B=\left[ {{M}^{1}}{{T}^{-2}}{{A}^{-1}} \right]
⇒z=[M1T−2A−1]\Rightarrow z=\left[ {{M}^{1}}{{T}^{-2}}{{A}^{-1}} \right]
x=3yz2x=3y{{z}^{2}}
⇒[M−1L−2T2Q2]=3y[M1T−2A−1]2\Rightarrow \left[ {{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}} \right]=3y{{\left[ {{M}^{1}}{{T}^{-2}}{{A}^{-1}} \right]}^{2}}
⇒y=[M1T−2A−1]2[M−1L−2T2Q2]\Rightarrow y=\dfrac{{{\left[ {{M}^{1}}{{T}^{-2}}{{A}^{-1}} \right]}^{2}}}{\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}} \right]}
⇒y=M−3L−2T4Q4\Rightarrow y={{M}^{-3}}{{L}^{-2}}{{T}^{4}}{{Q}^{4}}

So, the correct answer is Option A.

Additional Information:
Electromagnetic Induction is a current produced because of voltage production due to a changing magnetic field. This can happen when a conductor is placed in a moving magnetic field source is constantly moving in a stationary magnetic field.
The factors which affect magnetic induction are.
Number of Coils: The induced voltage is directly proportional to the number of coils of the wire. Greater the number of turns greater is the voltage produced.
Changing Magnetic Field: Changing magnetic field affects the induced voltage. This can be done by either moving the magnetic field around the conductor.

Note:
Some of the applications of electromagnetic induction are,
AC generator.
The working of electrical transformers.
The magnetic flow meter.