Question

The voltage V(in volts) that results if an alternating current I (in amps) flows through a resistor R is given thus: V = F(T) = IRsinωt. What is the maxima of the function?

Answer 1

Solution:

Given, $V = F(t) = IR\sinωt$

$V'= ωIR\cosωt \\ V''=-ω^2IR\sinωt$

Put V'=0

$\Rightarrow ωIR\cosωt=0 \\ \Rightarrow \cosωt=0 \\ \Rightarrow ωt=\frac{\pi }{2},\:ωt=\frac{3\pi }{2}; 0\le ωt\le 2\pi \\ \Rightarrow t=\frac{\pi }{2ω},\:ωt=\frac{3\pi }{2w}$

Put these values of t in V' '.

$V''=-ω^2IR\sin(ω\frac{\pi }{2ω})=-ω^2IR<0$

So, maxima exists.

Now, put $t=\frac{\pi }{2ω}$ in V.

$V= IR\sin(ω.\frac{\pi }{2ω}) \\ \Rightarrow V=IR(1) \\\Rightarrow V=IR$

This is maxima.