The current flowing through an inductor of self-inductance L is continuously increasing at constant rate. The variation of induced e.m.f. (e) verses $\text{dI}/\text{dt}$ is shown graphically by figure}
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Induced e.m.f. is zero if current is constant. If current increases at a constant rate, e.m.f. is a constant non-zero value.
Step 1: Understanding the Concept:
According to Faraday's Law of electromagnetic induction and Lenz's Law, when the current through an inductor changes, an electromotive force (e.m.f.) is induced across it that opposes the change in current. Step 2: Key Formula or Approach:
The formula for the induced e.m.f. ($e$) in a self-inductor is:
\[ e = -L \frac{dI}{dt} \]
where $L$ is the self-inductance (a positive constant characteristic of the coil), and $\frac{dI}{dt}$ is the rate of change of current. Step 3: Detailed Explanation:
We need to find the graphical relationship between $y = e$ and $x = \frac{dI}{dt}$.
The equation $e = -L \left( \frac{dI}{dt} \right)$ takes the mathematical form $y = -mx$, where the slope $m$ is $L$.
This equation describes a straight line passing through the origin.
Because the slope is negative (due to Lenz's law), as $\frac{dI}{dt}$ (the $x$-axis) is positive (current is increasing), the induced e.m.f. $e$ (the $y$-axis) will be negative.
Looking at the provided graphs:
- (A) shows a straight line through the origin entering the 4th quadrant (positive $x$, negative $y$). This perfectly matches our equation.
- (B) shows a straight line through the origin in the 1st quadrant, implying $e = +L(dI/dt)$, which ignores Lenz's Law.
- (C) shows a non-linear curve, which is incorrect.
- (D) shows a vertical shift or different dependency.
Therefore, graph (A) correctly represents the variation.
Checking the options, Option (B) points to graph "A". Step 4: Final Answer:
The variation is shown graphically by figure A, corresponding to option (B).
