The flux linked with a coil at any instant t is given by phi = 10t^2 - 50t + 250. The induced emf at

Q: Why is there a negative sign in Faraday's law formula?

The negative sign represents Lenz's Law. It physically means that the induced EMF (and resulting current) will always create a magnetic field that opposes the change in the original magnetic flux. It's nature's way of maintaining balance!

Q: What happens to the $+250$ when we calculate the EMF?

The $+250$ is a constant value. In calculus, the derivative of any constant is zero because a constant doesn't change with time. Since EMF only depends on the *change* in flux, the constant initial flux doesn't contribute to the induced voltage.

Q: Can an induced EMF really be negative?

Yes! In physics, a negative EMF doesn't mean "less than zero" in a purely mathematical sense; it indicates the *polarity* or direction of the induced voltage relative to the chosen positive direction of the coil.

Q: What would it mean if the calculated EMF was exactly zero?

If $e = 0$, it means the magnetic flux is momentarily constant (not changing). For our equation, this happens exactly at the vertex of the parabola, at $t = 2.5 \text{ s}$.

Q: I got $+10 \text{ V}$ instead of $-10 \text{ V}$. What did I do wrong?

You likely forgot the negative sign in Faraday's Law ($e = -d\phi/dt$). It's a very common silly mistake! Always write the formula with the negative sign first before substituting the derivative.

Question

The flux linked with a coil at any instant $t$ is given by $\phi = 10t^2 - 50t + 250$. The induced emf at $t = 3	ext{ s}$ is

Options:

(A) $-190	ext{ V}$

(B) $-10	ext{ V}$

(C) $10	ext{ V}$

(D) $190	ext{ V}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a conducting coil sitting in space. Suddenly, a magnetic field starts passing through it, and this field isn't constant—it's changing every second! In physics, the measure of this magnetic field passing through a given area is called **magnetic flux** ($\phi$).

In our problem, the flux is governed by a precise mathematical rule: 
$$ \phi = 10t^2 - 50t + 250 $$

This quadratic equation tells us exactly how much flux is linked with the coil at any given second $t$. But nature has a quirky response to changing flux. According to **Faraday's Law of Electromagnetic Induction**, whenever the magnetic flux through a coil changes, the coil fights back by generating its own voltage, known as the **induced EMF** ($e$).

### The Master Equation

Faraday's Law gives us the exact tool to calculate this induced voltage. It states that the induced EMF is the negative rate of change of magnetic flux with respect to time:
$$ e = -\frac{d\phi}{dt} $$

That negative sign is no accident! It represents **Lenz's Law**, which is essentially the universe's way of enforcing inertia. The induced EMF will always try to drive a current in a direction that *opposes* the very change in flux that created it.

### Executing the Calculus

To find the EMF, we need to differentiate our flux equation with respect to time $t$. Let's plug our expression into Faraday's formula:
$$ e = -\frac{d}{dt}(10t^2 - 50t + 250) $$

Now, we apply basic power rules of differentiation. The derivative of $10t^2$ is $20t$, the derivative of $-50t$ is $-50$, and the derivative of the constant $250$ is simply $0$. 
$$ e = -(20t - 50) $$

### Final Calculation

We have our general formula for the induced EMF at any time $t$. The question asks for the specific EMF at exactly $t = 3 	ext{ s}$. All we need to do is substitute $t = 3$ into our derived equation:
$$ e = -(20(3) - 50) $$
$$ e = -(60 - 50) $$
$$ e = -10 	ext{ V} $$

The induced EMF at $t = 3 	ext{ s}$ is **$-10 	ext{ V}$**. The negative value simply tells us the polarity of the induced voltage, confirming that it is actively opposing the increase in magnetic flux at that specific instant.

Animated Solution for Physics - Electromagnetic Induction: The flux linked with a coil at any instant $t$ is given by $ϕ = 10 t^{2} - 50 t + 250$ . The induced emf at $t = 3 s$ is

Select Answer:

Visualized Solution

Magnetic Flux

Faraday's Law

Substituting $ϕ$

Differentiation

EMF at $t = 3 s$

Significance of Negative Sign

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Analyzing the Setup

The Master Equation

Executing the Calculus

Final Calculation

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Animated Solution for Physics - Electromagnetic Induction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The flux linked with a coil at any instant t is given by ϕ=10t2−50t+250. The induced emf at t=3 s is