To find the electromotive force $E$ induced by the coil, we need to calculate the time derivative of the magnetic flux $\phi$. The expression for the flux is given as:

$\phi = K \sin(2 \pi f t)$

where:

• $K$ is a constant,
• $f$ is the frequency,
• $t$ is time.

Step 1: Compute the Derivative of $\phi$

The electromotive force $E$ is given by Faraday's Law:

$E = -N \frac{d\phi}{dt}$

To find $\frac{d\phi}{dt}$, let's differentiate $\phi = K \sin(2 \pi f t)$ with respect to $t$:

$\frac{d\phi}{dt} = \frac{d}{dt} \left( K \sin(2 \pi f t) \right)$

Applying the chain rule:

$\frac{d\phi}{dt} = K \cdot \frac{d}{dt} \left( \sin(2 \pi f t) \right)$

Since the derivative of $\sin(x)$ is $\cos(x)$, we have:

$\frac{d}{dt} \left( \sin(2 \pi f t) \right) = (2 \pi f) \cos(2 \pi f t)$

Therefore:

$\frac{d\phi}{dt} = K \cdot (2 \pi f) \cos(2 \pi f t)$

$\frac{d\phi}{dt} = 2 \pi f K \cos(2 \pi f t)$

Step 2: Determine the Induced EMF, $E$

Now substitute $\frac{d\phi}{dt}$ into Faraday's Law:

$E = -N \frac{d\phi}{dt}$

$E = -N (2 \pi f K \cos(2 \pi f t))$

Simplifying:

$E = -2 \pi N f K \cos(2 \pi f t)$

Final Answer

The induced electromotive force is:

$\boxed{E = -2 \pi N f K \cos(2 \pi f t)}$

This is the expression for the induced EMF in terms of the constants $N$, $f$, $K$, and time $t$.

Would you like more details, or do you have any questions?

Related Questions to Explore:

1. What happens to the EMF if the frequency $f$ is doubled?
2. How does the negative sign in Faraday's Law relate to Lenz's Law?
3. What is the physical significance of the constant $K$ in this context?
4. How would the expression for $E$ change if $\phi = K \cos(2 \pi f t)$ instead?
5. What are some practical applications of Faraday's Law in electrical engineering?

Tip: When dealing with sinusoidal functions, always pay attention to their derivatives, as they often involve phase shifts.