Answer
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Hint: The above problem can be resolved by using the concepts and applications of the dimensional formulas. The dimensional formula for the magnetic flux density can be obtained by the mathematical relation for the magnetic flux density. The magnetic flux density is determined by taking the ratio of the magnetic flux and the region's volume taken into consideration. Then the corresponding values are substituted, and the final result is obtained.
Complete Step by Step Solution:
A dimensional formula represents an equation, which gives the relation between fundamental units and derived units in terms of dimensions.
The length, mass and time are taken as three base dimensions and are represented by letters L, M, T respectively.
Magnetic flux is a measure of the quantity of magnetism, being the total number of magnetic lines of force passing through a specified area in a magnetic field. Magnetic flux through a plane of area $A$ placed in a uniform magnetic field $B$ can be written as ${\varphi _B} = B \cdot A = BA\cos \theta $.
The dimensional formula of area is $A = \left[ {{M^0}{L^2}{T^0}} \right]$ and
The dimensional formula of magnetic field is $B = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$ since, $B = \dfrac{{{\text{Force}}}}{{{\text{Charge} \times \text{Velocity}}}} = \dfrac{{\left[ {{M^1}{L^1}{T^{ - 2}}} \right]}}{{\left[ {{M^0}{L^0}{T^0}{I^1}} \right]\left[ {{L^1}{T^{ - 1}}} \right]}} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
Since $\cos \theta $ is a number, it has no dimensions.
Thus, the dimensional formula of magnetic flux is ${\varphi _B} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]\left[ {{L^2}} \right] = \left[ {{M^1}{L^2}{T^{ - 2}}{I^{ - 1}}} \right]$
Magnetic Flux Density is the amount of magnetic flux through unit area taken perpendicular to direction of magnetic flux. Mathematically, $b = \dfrac{{{\varphi _B}}}{A}$.
Thus, the dimensional formula of magnetic flux density is $b = \dfrac{{{\varphi _B}}}{A} = \dfrac{{\left[ {{M^1}{L^2}{T^{ - 2}}{I^{ - 1}}} \right]}}{{\left[ {{L^2}} \right]}} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
Note: Flux Density ($b$) is related to Magnetic Field ($B$) by $b = \mu B$ where $\mu $ is the permeability of the medium (material) where we are measuring the fields.
The permeability of the medium is a constant and has no dimensions.
Thus the dimensional formula of magnetic flux density is the same as that of the magnetic field $B$, which is given by, $B = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
Complete Step by Step Solution:
A dimensional formula represents an equation, which gives the relation between fundamental units and derived units in terms of dimensions.
The length, mass and time are taken as three base dimensions and are represented by letters L, M, T respectively.
Magnetic flux is a measure of the quantity of magnetism, being the total number of magnetic lines of force passing through a specified area in a magnetic field. Magnetic flux through a plane of area $A$ placed in a uniform magnetic field $B$ can be written as ${\varphi _B} = B \cdot A = BA\cos \theta $.
The dimensional formula of area is $A = \left[ {{M^0}{L^2}{T^0}} \right]$ and
The dimensional formula of magnetic field is $B = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$ since, $B = \dfrac{{{\text{Force}}}}{{{\text{Charge} \times \text{Velocity}}}} = \dfrac{{\left[ {{M^1}{L^1}{T^{ - 2}}} \right]}}{{\left[ {{M^0}{L^0}{T^0}{I^1}} \right]\left[ {{L^1}{T^{ - 1}}} \right]}} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
Since $\cos \theta $ is a number, it has no dimensions.
Thus, the dimensional formula of magnetic flux is ${\varphi _B} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]\left[ {{L^2}} \right] = \left[ {{M^1}{L^2}{T^{ - 2}}{I^{ - 1}}} \right]$
Magnetic Flux Density is the amount of magnetic flux through unit area taken perpendicular to direction of magnetic flux. Mathematically, $b = \dfrac{{{\varphi _B}}}{A}$.
Thus, the dimensional formula of magnetic flux density is $b = \dfrac{{{\varphi _B}}}{A} = \dfrac{{\left[ {{M^1}{L^2}{T^{ - 2}}{I^{ - 1}}} \right]}}{{\left[ {{L^2}} \right]}} = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
Note: Flux Density ($b$) is related to Magnetic Field ($B$) by $b = \mu B$ where $\mu $ is the permeability of the medium (material) where we are measuring the fields.
The permeability of the medium is a constant and has no dimensions.
Thus the dimensional formula of magnetic flux density is the same as that of the magnetic field $B$, which is given by, $B = \left[ {{M^1}{T^{ - 2}}{I^{ - 1}}} \right]$.
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