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A total charge Q flows across a resistor...

A total charge Q flows across a resistor R during a time interval T in such a way that the current versus time graph for 0 to T is like the loop of a sine curve in the range `0 "to" pi`. The total heat generated in the resistor is

A

`Q^2pi^2R//8T`

B

`2Q^2pi^2R//T`

C

`Q^2pi^2R//T`

D

`Q^2pi^2R//2T`

Text Solution

Verified by Experts

The correct Answer is:
A
a. Time period: `T_0 = 2T, omega = (2pi)/T_0 = pi/T`
`I = I_0 sin omegat = I_0 sin (pi/T t)`
`Q = int_(0)^(T) I dt`
`rArr = I_(0) int_(0)^(T) sin (pi/T t) dt `
`= (-I_0)/(pi//T) [ cos (pi/T t)]_0 = (2TI_0)/pi`
or `I_0 = Qpi//2T`
Heat `= int_(0)^(T) I^2 R dt = int_(0)^(T) I_(0)^(2) sin^2 (pi/T t) R dt = Q^2pi^2R//8T`.
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