Question

The dimensional formula for wavelength is equivalent to (Where E is the electric field, B is the magnetic field, L is Induction and C is Capacitance):
\[{\text{A}}{\text{.}}\dfrac{{E\sqrt {LC} }}{B}\]
\[{\text{B}}{\text{. }}\dfrac{E}{{B\sqrt {LC} }}\]
\[{\text{C}}{\text{.}}\dfrac{{B\sqrt {LC} }}{E}\]
\[{\text{D}}{\text{. }}\dfrac{B}{{E\sqrt {LC} }}\]

Answer 1

Hint: For solving these types of questions we must remember the definition and dimensional formula of wavelength, electric field, magnetic field, induction and capacitance.

Complete step-by-step answer:
Wavelength is the distance between two points on the wave at the same time and on successive waves. For example- two straight troughs, or crests.

The Dimension formula of wavelength is $[{M^0}{L^1}{T^0}]$.
Electric field is defined as the field that surrounds the electrical charge and it exerts attracting or repelling force to the other charges in the field. Electric fields are created by electrical charges or by magnetic fields varying in time.

Dimensional formula of electric field, E is $[{M^1}{L^1}{I^{ - 1}}{T^{ - 3}}]$
Magnetic field is a vector field that defines the magnetic effect in relative motion of electrical charges and magnetic materials. A charge traveling parallel to a current of other charges encounters a force that is perpendicular to its own direction.

Its dimensional formula is B, $[{M^1}{T^{ - 2}}{A^{ - 1}}]$
Electromagnetic or magnetic induction is the creation of an electromotive force in a changing magnetic field over an electrical conductor.
Dimensional formula of induction is E, $[{M^1}{L^2}{T^{ - 2}}{A^{ - 2}}]$
Capacitance is defined as the ratio of the electric charge on each conductor to the potential difference between them.
Dimensional formula of capacitance is C, $[{M^{ - 1}}{L^{ - 2}}{T^4}{I^2}]$

Therefore, we have
L=$[{M^1}{L^2}{T^{ - 2}}{A^{ - 2}}]$
C=$[{M^{ - 1}}{L^{ - 2}}{T^4}{A^2}]$
B=$[{M^1}{T^{ - 2}}{A^{ - 1}}]$
E= $[{M^1}{L^1}{T^{ - 3}}{A^{ - 1}}]$
Where $M$ is for mass, $L$ is for length, $T$ is for time and $A$ is for current.

Now we will use the above given expressions to derive our final equation
In all the options \[\sqrt {LC} \] is common. Hence, we will find the value of \[\sqrt {LC} \] using the above given dimensions
\[\sqrt {LC} \]=\[\sqrt {{M^1}{L^2}{T^{ - 2}}{A^{ - 2}} \times {M^{ - 1}}{L^{ - 2}}{T^4}{A^2}} \]=\[\sqrt {{M^0}{L^0}{T^2}{A^0}} \]=\[\sqrt {{T^2}} \]=$T$
Also, we can see that \[\dfrac{B}{E}\] or \[\dfrac{E}{B}\] , hence we will find the value of \[\dfrac{B}{E}\]
\[\dfrac{B}{E}\]=$\dfrac{{{M^1}{T^{ - 2}}{A^{ - 1}}}}{{{M^1}{L^1}{T^{ - 3}}{A^{ - 1}}}} = {M^0}{L^{ - 1}}{T^1}{A^0} = {L^{ - 1}}{T^1}$

Now we will enter the value of \[\sqrt {LC} \]= $T$----(i)
and \[\dfrac{B}{E}\]=\[{L^{ - 1}}{T^1}\] ---(ii)
In our given options
\[{\text{A}}{\text{.}}\dfrac{{E\sqrt {LC} }}{B}\] ,\[{\text{B}}{\text{. }}\dfrac{E}{{B\sqrt {LC} }}\], \[{\text{C}}{\text{.}}\dfrac{{B\sqrt {LC} }}{E}\], \[{\text{D}}{\text{. }}\dfrac{B}{{E\sqrt {LC} }}\]

\[{\text{A}}{\text{.}}\dfrac{{E\sqrt {LC} }}{B} = {({L^{ - 1}}{T^1})^{ - 1}} \times T = {L^1}{T^0} = {M^0}{L^1}{T^0}\]
\[{\text{B}}{\text{. }}\dfrac{E}{{B\sqrt {LC} }} = {({L^{ - 1}}{T^1})^{ - 1}} \times {T^{ - 1}} = {L^{ - 1}}{T^{ - 2}} = {M^0}{L^{ - 1}}{T^{ - 2}}\],
 \[{\text{C}}{\text{.}}\dfrac{{B\sqrt {LC} }}{E} = ({L^{ - 1}}{T^1}) \times T = {L^{ - 1}}{T^2} = {M^0}{L^{ - 1}}{T^2}\],
\[{\text{D}}{\text{. }}\dfrac{B}{{E\sqrt {LC} }} = ({L^{ - 1}}{T^1}) \times {T^{ - 1}} = {L^{ - 1}}{T^0} = {M^0}{L^{ - 1}}{T^0}\]
All the values have been taken from (i) and (ii)
We know the dimensional formula of wavelength is $[{M^0}{L^1}{T^0}]$, hence option A matches our dimension. Therefore, option A is the correct answer and dimensional formula for wavelength is equivalent to \[\dfrac{{E\sqrt {LC} }}{B}\].

Note: In our question we saw that wavelength, electric field, magnetic field, Induction and capacitance are very important parts of physics and their dimensional formula is very important. Hence, we must always remember their properties and formulas.