Question

The magnetic flux linked with a coil, in webers, is given by the equation \(\Phi=4 t^{2}-t+7\). Then the magnitude of induced emf at 2 sec will be ... (a) \(15 \mathrm{v}\) (b) \(19 \mathrm{v}\) (c) \(17 \mathrm{v}\) (d) \(21 \mathrm{v}\)

Short answer

The magnitude of the induced emf at 2 seconds is \(15 V\).

Step-by-step solution

Differentiate the given magnetic flux equation with respect to time

To find the rate of change of magnetic flux, differentiate the given equation with respect to time (t): Given equation: \(\Phi=4t^2 - t + 7\) Now, differentiate it with respect to 't': \(\frac{d\Phi}{dt} = \frac{d}{dt}(4t^2 - t + 7)\) Applying the differentiation, we get: \(\frac{d\Phi}{dt} = 8t - 1\)

Evaluate the expression at t = 2 sec

Substitute t = 2 sec in the equation from Step 1: \(\frac{d\Phi}{dt} = 8(2) - 1\) Calculating the value, we get: \(\frac{d\Phi}{dt} = 16 - 1\) \(\frac{d\Phi}{dt} = 15\)

Apply Faraday's Law to find the induced emf

According to Faraday's Law, the induced emf (E) is given by: \(E = -\frac{d\Phi}{dt}\) We found that \(\frac{d\Phi}{dt} = 15\) in Step 2, so the induced emf (E) can be calculated as follows: \(E = -15\)

Find the magnitude of the induced emf

The magnitude of induced emf is the absolute value of E obtained in Step 3: \(Magnitude\: of\: E = |-15|\) Calculating the magnitude, we get: \(Magnitude\: of\: E = 15 V\) Therefore, at 2 seconds, the magnitude of the induced emf is 15 V. The correct answer is (a) \(15 V\).

Key concepts

Magnetic Flux

Magnetic flux is a key concept in electromagnetism that signifies the quantity of magnetic field passing through a given surface. It helps us understand how magnetic fields interact with different materials and can be visualized as the number of magnetic field lines that pass through a surface. Mathematical representation of magnetic flux, typically denoted by the symbol \( \Phi \), is calculated based on both the strength of the magnetic field and the area through which it penetrates, in a direction perpendicular to the field lines.
In equations, magnetic flux is often expressed as:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

Here:

• \( B \) represents the magnetic field strength (in teslas).
• \( A \) is the area of the surface through which the field lines pass (in square meters).
• \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.

Understanding this concept is essential for studying how changing magnetic fields can induce electric currents in coils or loops.

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. It is an essential tool in calculus and is widely applied in physics to analyze changes over time. For example, in the given exercise, differentiation is used to determine the rate of change of the magnetic flux \( \Phi \) with respect to time \( t \). To differentiate a function, follow these steps:

• Identify the function you need to differentiate, such as \( f(t) = 4t^2 - t + 7 \).
• Apply differentiation rules to find the function’s derivative, which provides the rate of change. In our case, differentiate by applying the power rule: for every term \( at^n \), it becomes \( nant^{n-1} \).

Let's differentiate the function:

1. For \( 4t^2 \), it becomes \( 8t \).
2. For \( -t \), it becomes \( -1 \).
3. For \( +7 \), it becomes \( 0 \) (since constants vanish through differentiation).

Thus, the derivative \( \frac{d\Phi}{dt} = 8t - 1 \) describes how the magnetic flux changes with time. In physics problems, this is often the first step in finding the induced electromotive force (emf).

Faraday's Law

Faraday's Law is one of the fundamental principles of electromagnetism, explaining how a changing magnetic field induces an electromotive force (emf) in a conductor. This law is crucial for understanding how electric generators, transformers, and inductors work. According to Faraday's Law, the induced emf (\( E \)) in a closed loop is directly proportional to the rate of change of magnetic flux through the loop.Mathematically, Faraday’s Law is expressed as:

• \( E = -\frac{d\Phi}{dt} \)

The negative sign in the formula indicates that the induced emf generates a current that opposes the change in magnetic flux, aligning with Lenz's law. Applying Faraday's Law to solve the exercise, you determine the rate of change of magnetic flux from differentiation, \( \frac{d\Phi}{dt} = 15 \), and thus the magnitude of the induced emf becomes |

• \( E = |-\frac{d\Phi}{dt}| = 15 \) Volts.

Faraday's Law not only helps in calculating the amount of induced emf but also forms the basis for understanding how electric circuits respond to changes in magnetic environments.