Alternating Current

We need to find:
For the power dissipated to become half of its maximum value, what is the current amplitude?
Step 1: Relation between Power and Current
The average power dissipated in an AC circuit is given by:
P=Irms2R
At resonance, the impedance is purely resistive ( Z=R ), and the current is maximum ( I=I‌max ‌ ).
So maximum power is:
Pmax=Irms,max2‌‌R=(‌

Imax

√2

)2R=‌

Imax2‌‌R

2

But we only need relative power, so we can use P∝I2.
Step 2: Condition for half power
We are told P=‌

1

2

P‌max ‌
So:
I2R=‌

1

2

Imax2R
Simplify:
I=‌

Imax

√2

So the current amplitude becomes 1∕√2 times its maximum value when the power is half of the maximum.