Induced EMF Calculation from Magnetic Flux

The problem asks us to determine the induced EMF in a single turn coil at a particular instant in time, given the equation for the magnetic flux ($\Phi$) linked with the coil as a function of time ($t$).

To solve this, we will use Faraday's Law of Electromagnetic Induction, which establishes the relationship between a changing magnetic flux and the induced electromotive force (EMF).

Magnetic Flux and Faraday's Law Principle

• The magnetic flux ($\Phi$) linked with the coil at an instant of time $t$ is provided by the equation:
• $$\Phi(t) = 2t^2 - 20t + 40$$
• According to Faraday's Law of Electromagnetic Induction, the magnitude of the induced EMF ($\mathcal{E}$) in a coil is equal to the negative rate of change of magnetic flux with respect to time. This is expressed mathematically as:
• $$\mathcal{E} = -\frac{d\Phi}{dt}$$
• Here, $\frac{d\Phi}{dt}$ represents the derivative of the magnetic flux function with respect to time, indicating how quickly the magnetic flux is changing.

Step-by-Step Induced EMF Determination

Let's follow these steps to find the induced EMF at $t = 2$ seconds:

1. Differentiate the Magnetic Flux Function:

• We are given the magnetic flux function: $$\Phi(t) = 2t^2 - 20t + 40$$
• To find the rate of change of magnetic flux, we differentiate $\Phi(t)$ with respect to $t$:
• $$\frac{d\Phi}{dt} = \frac{d}{dt}(2t^2 - 20t + 40)$$
• Applying the rules of differentiation (power rule and constant multiple rule):
• The derivative of $2t^2$ is $2 \cdot (2t^{2-1}) = 4t$.
• The derivative of $-20t$ is $-20 \cdot (1t^{1-1}) = -20$.
• The derivative of a constant ($40$) is $0$.
• So, the derivative of the magnetic flux is:
• $$\frac{d\Phi}{dt} = 4t - 20$$

2. Apply Faraday's Law to Calculate EMF:

• Now, we use Faraday's Law, $\mathcal{E} = -\frac{d\Phi}{dt}$:
• Substitute the expression for $\frac{d\Phi}{dt}$ into the EMF formula:
• $$\mathcal{E} = -(4t - 20)$$
• Distribute the negative sign:
• $$\mathcal{E} = -4t + 20$$
• This equation gives the induced EMF at any instant $t$.

3. Calculate EMF at the Specific Time:

• The problem asks for the induced EMF at the instant $t = 2$ seconds.
• Substitute $t = 2$ into the EMF equation we just derived:
• $$\mathcal{E}(t=2) = -4(2) + 20$$
• Perform the multiplication:
• $$\mathcal{E}(t=2) = -8 + 20$$
• Perform the addition:
• $$\mathcal{E}(t=2) = 12 \, \text{V}$$

Therefore, the induced EMF in the coil at the instant $t = 2$ seconds is 12 Volts.