Q. 13.9( 9 Votes )

The amount of charge passed in time t through a cross-section of a wire is Q(t) = At2 + Bt + C.

(a) Write the dimensional formulae for A, B and C.

(b) If the numerical values of A, B and C are 5, 3 and 1 respectively in S.I. units, find the value of the current at t = 5 s.

Answer :

a) IT-1, I, IT b)53A

Given,


Charge as a function of time is Q(t) = At2 + Bt + C.


The principle of homogeneity states that each term on the either side of an equation has the same dimensions.


a) Each terms on the Right Hand Side of the equation has the same unit, and hence the dimension of that of the term on the Left Hand Side.


So, each term on RHS is having same dimensions as of the quantity Charge, Q.


We know that, Charge Q is



Where I is current with dimension ‘I’ and t is time in seconds with dimension ‘T’.


Hence the dimension of Q or Q(t)is ‘IT’.


By inspection, we can see that the term C in RHS is devoid of any other quantities and hence C also has the dimension ‘IT’ (Ans.)


We know that dimension of the term At2 is also ‘IT’, and t represents time (Dimension T).


So,


Or



Similarly,



Or,



So dimensions of A, B, and C are IT-1, I, IT respectively.


b) The expression for the charge at time t can be rewritten by assigning values to the constants as.



We know that instantaneous current, I can be expressed as



By substituting the given expression in the above equation, we get,



Or,



For t=5s, I becomes


Hence the current at t=5s is 53A


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