Problem 18 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The magnetic flux through each loop of a 75 -loop coil is given by \(\left(8.8 t-0.51 t^{3}\right) \times 10^{-2} \mathrm{~T} \cdot \mathrm{m}^{2},\) where the time \(t\) is in seconds. (a) Determine the emf \(\mathscr{E}\) as a function of time. (b) What is \(\mathscr{E}\) at \(t=1.0 \mathrm{~s}\) and \(t=4.0 \mathrm{~s} ?\)

Step-by-Step Solution

Answer

(a) \( \mathscr{E}(t) = -0.75 \left(8.8 - 1.53t^2 \right) \). (b) \( \mathscr{E}(1.0) = -5.45 \text{ V}, \mathscr{E}(4.0) = 11.76 \text{ V}.\)

Step 1: Recognize the Formula for EMF

The electromotive force (EMF) \( \mathscr{E} \) induced in a coil is given by Faraday's Law: \( \mathscr{E} = - N \frac{d\Phi}{dt} \), where \( N \) is the number of loops and \( \Phi \) is the magnetic flux through one loop.

Step 2: Differentiate the Magnetic Flux Function

The magnetic flux \( \Phi(t) \) is given by \( \Phi(t) = (8.8t - 0.51t^3) \times 10^{-2} \). Differentiate \( \Phi(t) \) with respect to \( t \) to find \( \frac{d\Phi}{dt} \). The differentiation gives: \( \frac{d\Phi}{dt} = \left( 8.8 - 3 \times 0.51t^2 \right) \times 10^{-2} \).

Step 3: Calculate EMF as a Function of Time

Using \( \mathscr{E} = - N \frac{d\Phi}{dt} \), substitute \( N = 75 \) and \( \frac{d\Phi}{dt} = \left( 8.8 - 1.53t^2 \right) \times 10^{-2} \). Thus, \( \mathscr{E}(t) = - 75 \times \left( 8.8 - 1.53t^2 \right) \times 10^{-2} \). Simplify this to get \( \mathscr{E}(t) = - 0.75 \left( 8.8 - 1.53t^2 \right) \).

Step 4: Calculate EMF at Specific Time Points

Substitute \( t = 1.0 \) and \( t = 4.0 \) into the expression for \( \mathscr{E}(t) \).\[ \mathscr{E}(1.0) = - 0.75 \times (8.8 - 1.53 \times 1^2) = -0.75 \times 7.27 = -5.45 \text{ V}. \] \[ \mathscr{E}(4.0) = - 0.75 \times (8.8 - 1.53 \times 4^2) = -0.75 \times (-15.68) = 11.76 \text{ V}. \]

Key Concepts

Magnetic Flux

Magnetic flux is a fundamental concept in electromagnetism and is defined as the total magnetic field that passes through a given area. Imagine magnetic field lines passing through a surface; the magnetic flux represents the quantity of these field lines. It is calculated by the formula \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area through which the magnetic field lines pass, and \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.
Understanding magnetic flux is essential when dealing with electromagnetic phenomena such as induction. Magnetic flux can change over time, especially if the magnetic field or the area it interacts with changes. The greater the magnetic flux, the more field lines pass through the surface. This change is crucial because it leads to the generation of electromotive force (EMF), as described by Faraday's law. In our exercise, the magnetic flux through each loop of the coil changes with time, which in turn affects the resulting EMF.

Electromotive Force (EMF)

Electromotive force (EMF) is a term that describes the voltage or potential difference generated by a source of electrical energy. In the context of electromagnetic induction, EMF is induced in a circuit due to the changing magnetic flux passing through it. Faraday's Law of Induction provides the foundational principle for understanding how EMF is produced.
The law states \( \mathscr{E} = -N \frac{d\Phi}{dt} \), where \( \mathscr{E} \) is the induced EMF, \( N \) is the number of loops in the coil, and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux. The negative sign signifies Lenz's Law, indicating that the induced EMF opposes the change in flux. This principle assures that the EMF attempts to maintain the status quo by resisting the magnetic change. In the exercise, we calculate EMF over time by considering how magnetic flux changes with time. As a result, we derive an expression for EMF by substituting these variables into Faraday's formula.

Differentiation in Physics

Differentiation is a mathematical process and an essential tool in physics for analyzing how a quantity changes with respect to another. It provides insight into rates of change, which is fundamentally what drives physical laws like that of induction. By differentiating, we measure how one variable changes as another one varies.
In the exercise, differentiation is used to understand how magnetic flux changes with time. Since magnetic flux \( \Phi(t) = (8.8t - 0.51t^3) \times 10^{-2} \), differentiating this expression with respect to time \( t \) gives us its rate of change with time, \( \frac{d\Phi}{dt} \). The differentiation involves straightforward calculus:

• The derivative of \( 8.8t \) is \( 8.8 \).
• The derivative of \( -0.51t^3 \) is \( -3 \times 0.51t^2 \).

Performing these steps gives \( \frac{d\Phi}{dt} = (8.8 - 1.53t^2) \times 10^{-2} \). This result is crucial for calculating EMF using Faraday's Law. Understanding differentiation allows us to process functions that express dynamics within physical systems.