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Question

The square root of the product of inductance and capacitance has the dimension of


A

length

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B

mass

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C

time

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D

No dimension

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Solution

The correct option is C

time


Step 1: Given Data:

Let the capacitance be C and inductance be L

It is given that the square root of the product of inductance and capacitance i.e.,LC

Step 2: Formula Used:

The dimension of capacitance is C=M-1L-2T4A2

We know that, C=QV1 where QandVare charge and potential respectively.

Dimension of Q=AT2

Dimension of potential V is given by-

V=WQ3

Work has the dimension

W=F×d=MLT-2×L=ML2T-24

then the dimension of potential, using equations (2) and (4)

V=WQ=ML2T-2AT=ML2I-1T-35

Now the dimension of capacitance has been calculated as, using equations (2) and (5)

C=QV=ATML2T-3A-1=M-1L-2T4A26

The dimension of inductance L=ML2T-2A-2

We know that, L=VdIdt where VanddIdtare potential and rate of flow of current respectively.

Using equation (3)

Dimension of V=ML2A-1T-3

Dimension of dIdt=AT-1

Now for the dimension of inductance,

L=VdIdt=ML2T-3A-1AT-1=ML2T-2A-2

The dimension of inductance is L=ML2T-2A-27

Step 3: Calculation of the dimension of LC

According to the question, it is required to find the dimension of the square root of the product of inductance and capacitance i.e.,LC

It can be calculated using the equation (6) and (7)

=LC=ML2T-2A-2M-1L-2T4A2=T2=T

Hence option C is the correct answer.


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