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Q.
The current $i$ in a coil varies with time as shown in the figure. The variation of induced emf with time would be
NTA AbhyasNTA Abhyas 2022
Solution:
Induced emf, $\text{e} = - \text{L} \frac{\text{di}}{\text{dt}}$
For $0 \leq \text{t} \leq \frac{\text{T}}{4}$ ,
i-t graph is a straight line with positive constant slope.
$∴ \, \, \frac{\text{di}}{\text{dt}} = \text{constant}$
$⇒ \, \, \text{e} = - \text{ve}$ and constant For $0 \leq \text{t} \leq \frac{\text{T}}{4}$
For $\frac{\text{T}}{4} \leq \text{t} \leq \frac{\text{T}}{2} ,$
$∵ \, \, $ i is constant $∴ \, \, \frac{\text{di}}{\text{dt}} = 0$
$⇒ \, \, \text{e} = 0$ For $\frac{\text{T}}{4} \leq \text{t} \leq \frac{\text{T}}{2}$
For $\frac{\text{T}}{2} \leq \text{t} \leq \frac{3 \text{T}}{4}$ ,
i-t graph is a straight line with negative constant slope.
$∴ \, \, \frac{\text{di}}{\text{dt}} = \text{constant}$
⇒ e=+ve and constant For $\frac{\text{T}}{2} \leq \text{t} \leq \frac{3 \text{T}}{4}$
For $\frac{3 \text{T}}{4} \leq \text{t} \leq \text{T}$ ,
i is zero $∴ \, \, \frac{\text{di}}{\text{dt}} = 0$
$⇒ \, \, \text{e} = 0$ For $\frac{3 \text{T}}{4} \leq \text{t} \leq \text{T}$
From this analysis, the variation of induced emf with time as shown in the figure below.
