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Question: The correct relation between.\(B,H\text{ and }I\).is \(\begin{aligned} & \text{A}\text{. }B={{...

The correct relation between.B,H and IB,H\text{ and }I.is
A. B=μ0I×H B. B=μ0IH C. B=μ0(I+H) D. B=μ0IH \begin{aligned} & \text{A}\text{. }B={{\mu }_{0}}I\times H \\\ & \text{B}\text{. }B=\dfrac{{{\mu }_{0}}I}{H} \\\ & \text{C}\text{. }B={{\mu }_{0}}\left( I+H \right) \\\ & \text{D}\text{. }B={{\mu }_{0}}I-H \\\ \end{aligned}

Explanation

Solution

BB represents the magnetic induction, HH represents the magnetizing field intensity, and II represents the intensity of magnetization, for a given system. II is also represented as MM. They are all interrelated to each other. Write the expression for each of them and do some substitution to get the relationship between B,H and IB,H\text{ and }I.

Formula used:
B=μ0ni H=ni M or I=magnetic moment developedVolume Magnetic moment developed,  m= pole strength developed×distance between the poles \begin{aligned} & B={{\mu }_{0}}ni \\\ & H=ni \\\ & M\text{ or }I=\dfrac{\text{magnetic moment developed}}{\text{Volume}} \\\ & \text{Magnetic moment developed, } \\\ & m=\text{ pole strength developed}\times \text{distance between the poles} \\\ \end{aligned}
Where,
μ0=permeability n=number or rotation of a solenoid. i= current through the solenoid. \begin{aligned} & {{\mu }_{0}}=\text{permeability} \\\ & \text{n=number or rotation of a solenoid}\text{.} \\\ & i=\text{ current through the solenoid}\text{.} \\\ \end{aligned}

Complete step by step answer:
Magnetic induction (BB): When current flows inside the wire of a solenoid some magnetic moments developed. This current ii induces some magnetic field inside solenoid called magnetic induction and is given by B=μ0niB={{\mu }_{0}}ni, where n= number of turns of solenoidn=\text{ number of turns of solenoid}.

Magnetizing field intensity (HH): The ability of the magnetizing field to magnetize a material medium is called magnetic field intensity. Its magnitude is defined as the number of turns of solenoid per unit length required to produce a given magnetic field,
So H=niH=ni, but B0=μ0ni=μ0H{{B}_{0}}={{\mu }_{0}}ni={{\mu }_{0}}H
The intensity of magnetization (II).When a magnetic material is placed in the magnetizing field it gets magnetized. The magnetic moment developed per unit volume of the material is called intensity of magnetization, thus M=mVM=\dfrac{m}{V}
If magnetic induction Bm{{B}_{m}} produced due to magnetizing field then magnetic field intensity.
I=μ0MI={{\mu }_{0}}M
As the total magnetic field or the magnetic induction B\overrightarrow{B}inside a magnetic material is the resultant of magnetizing field Bo\overrightarrow{{{B}_{o}}}and the field Bm\overrightarrow{{{B}_{m}}}produced due to magnetization of the material. So

B=B0+Bm=μ0H+μ0I B=μ0(H+I) \begin{aligned} & B={{B}_{0}}+{{B}_{m}}={{\mu }_{0}}H+{{\mu }_{0}}I \\\ & \Rightarrow B={{\mu }_{0}}(H+I) \\\ \end{aligned}

Hence, the correct answer is option C.

Note:
Note that the S.I unit of magnetic induction is tesla(T) or weber×meter2(Wbm2)tesla(T)\text{ or }weber\times mete{{r}^{-2}}(Wb{{m}^{-2}}) which is equivalent to Nm1A1 or JouleA1m2N{{m}^{-1}}{{A}^{-1}}\text{ or }Joule{{A}^{-1}}{{m}^{-2}}. And the S.I unit of Magnetizing field intensity is
Ampere×meterr1Ampere\times meter{{r}^{-1}} which is equivalent to Nm2T or JouleWeber×meter\dfrac{N}{{{m}^{2}}T}\text{ or }\dfrac{Joule}{Weber\times meter}.