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Alternating Current — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsAlternating Current5 Marks
Question
For an LCR circuit driven at frequency $\omega$, the equation reads $\text{L}\frac{\text{di}}{\text{dt}}+\text{Ri}+\frac{\text{q}}{\text{C}}=\text{v}_\text{i}=\text{v}_\text{m}\sin\omega\text{t}$.
  1. Multiply the equation by i and simplify where possible.
  2. Interpret each term physically.
  3. Cast the equation in the form of a conservation of energy statement.
  4. Intergrate the equation over one cycle to find that the phase difference between v and i must be acute.
✓

Answer

Applying KVL for the loop as shown in the figure, we can write

$\text{V}=\text{V}_\text{m}\sin\omega\text{t}$
$\Rightarrow\ \text{L}\frac{\text{di}}{\text{dt}}=\frac{\text{q}}{\text{C}}+\text{iR}=\text{V}_\text{m}\sin\omega\text{t}\ .....\text{(i)}$
Multiplying both sides by i, we get
$\text{L}\frac{\text{di}}{\text{dt}}=\frac{\text{q}}{\text{C}}\text{i}+\text{i}^2\text{R}=(\text{V}_\text{m}\text{i})\sin\omega\text{t}=\text{Vi}\ .....\text{(ii)}$
where $\text{Li}\frac{\text{di}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\Big(\frac{1}{2}\text{Li}^2\Big)=$ rate of change of energy stored in an inductor.
Now, power loss in from of heat is given $i^2R$
$\frac{\text{q}}{\text{C}}\text{i}=\frac{\text{d}}{\text{dt}}\Big(\frac{\text{q}^2}{2\text{C}}\Big)=$ rate of chage of energy stored in the capacitor.
Vi = rate at which driving force pours in energy. It goes into (i) ohmic loss and (ii) increase of stored energy.
Hence Eq. (ii) is in the form of conservation of energy statement. Integrating both sides of Eq. (ii) with respect to time over one full cycle (0 → T) we may write
$\int_0^\text{T}\frac{\text{d}}{\text{dt}}\bigg(\frac{1}{2}\text{Li}^2+\frac{\text{q}^2}{2\text{C}}\bigg)\text{dt}+\int_0^\text{T}\text{Ri}^2\text{dt}=\int_0^\text{T}\text{Vi dt}$
$\Rightarrow\ 0+(+\text{ve})=\int_0^\text{T}\text{Vi dt}$
$\Rightarrow\ \int_0^\text{T}\text{Vi dt}>0$ if phase difference between V and i is a constant and acute angle.

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