Q. 13.9( 9 Votes )

# The amount of charge passed in time t through a cross-section of a wire is Q(t) = At^{2} + 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) = At^{2} + 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 At^{2} 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

Rate this question :

An inductance of 2.0 H, a capacitance of 18 μF and a resistance of 10 kΩ are connected to an AC source of 20 V with adjustable frequency.

(a) What frequency should be chosen to maximize the current in the circuit?

(b) What is the value of this maximum current?

HC Verma - Concepts of Physics Part 2

A cell of emf E and internal resistance r is connected across an external resistance R. Plot a graph showing the variation of P.D. across R, verses R.

Physics - ExemplarTwo non-ideal batteries are connected in series. Consider the following statements:

(A) The equivalent emf is larger than either of the two emfs.

(B) The equivalent internal resistance is smaller than either of the two internal resistances.

HC Verma - Concepts of Physics Part 2

Does a conductor become charged when a current is passed through it?

HC Verma - Concepts of Physics Part 2