### Explanation of Faraday’s Electromagnetic Law

Michael Faraday, an English scientist, made one of the most significant discoveries in the field of electromagnetism in the early 19th century. Faraday’s Law of Electromagnetic Induction, often referred to as Faraday’s Law, relates to the generation of an electromotive force (EMF) in a coil due to a change in magnetic flux through the coil. This fundamental principle has applications in various modern technologies, including electrical generators, transformers, and inductive charging systems.

Faraday’s Law can be succinctly stated: the EMF induced in a closed circuit is directly proportional to the rate of change of the magnetic flux through the circuit. In mathematical terms, this relationship can be expressed as:

\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]

where:

– \(\mathcal{E}\) is the induced electromotive force in volts (V)

– \(d\Phi_B/dt\) is the rate of change of magnetic flux in webers per second (Wb/s)

The negative sign indicates the direction of the induced EMF and current, determined by Lenz’s Law, which states that the induced current will flow in a direction that opposes the change in flux that produced it. This is a statement of conservation of energy.

Magnetic flux (\(\Phi_B\)) is defined as the product of the strength of the magnetic field (B), the area through which the field lines pass (A), and the cosine of the angle (\(\theta\)) between the magnetic field lines and the perpendicular to the surface (A):

\[ \Phi_B = BA\cos(\theta) \]

This law has profound implications and can lead to intricate problems depending on the complexity of the magnetic fields and the motion of conductors within these fields. Here are some sample problems and solutions related to Faraday’s Law to illustrate its application.

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### Problems and Solutions related to Faraday’s Electromagnetic Law

**Problem 1**: A loop of wire with an area of \(0.02 m^2\) is in a magnetic field of \(0.5 T\). The magnetic field is decreased to \(0.1 T\) over a time of \(0.3 s\). Find the magnitude of the induced EMF in the loop.

**Solution**:

Using Faraday’s Law, the induced EMF is

\[ \mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t} = -\frac{B_fA – B_iA}{\Delta t} \]

where \(B_f\) is the final magnetic field, \(B_i\) is the initial magnetic field, \(A\) is the area, and \(\Delta t\) is the change in time.

Plugging the given values into the equation, we get:

\[ \mathcal{E} = -\frac{0.1 \cdot 0.02 – 0.5 \cdot 0.02}{0.3} = -\frac{(-0.008)}{0.3} \approx 0.0267 V \]

**Problem 2**: A square loop of wire with side length \(0.1 m\) is perpendicular to a magnetic field of \(1 T\). If the loop enters a region where the field is \(0 T\) at a speed of \(1 m/s\), calculate the induced EMF in the loop as it exits the region.

**Solution**:

Firstly, calculate the change in flux. The initial flux when in the \(1 T\) field is

\[ \Phi_i = BA = 1 \cdot (0.1)^2 = 0.01 Wb \]

As the loop exits the region, the final flux is \(0 Wb\). The induced EMF, as the loop exits the field, can be calculated using the equation

\[ \mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t} \]

The time to exit the region can be found by dividing the length of the side by the speed.

\[ \Delta t = \frac{A}{v} = \frac{0.1}{1} = 0.1 s \]

Thus,

\[ \mathcal{E} = -\frac{0 – 0.01}{0.1} = 0.1 V \]

**Problem 3**: A coil of wire with 100 turns and an area of \(0.01 m^2\) is oriented parallel to a magnetic field of \(0.02 T\). If the coil is rotated to be perpendicular to the magnetic field in \(0.2 s\), find the average induced EMF.

**Solution**:

Calculate the initial and final flux. Initially, \(\Phi_i = 0\) because the coil is parallel to the field (\(\theta = 0^\circ\), and \(\cos(0^\circ) = 1\)), and the final flux is

\[ \Phi_f = NBA\cos(\theta) = 100 \cdot 0.02 \cdot 0.01 \cdot 1 = 0.02 Wb \]

The average EMF is

\[ \mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t} = -\frac{0.02 – 0}{0.2} \cdot 100 = -10 V \]

**Problem 4**: A conducting rod of length \(l = 0.5m\) moves at constant velocity \(v = 2m/s\) perpendicular to a constant magnetic field \(B = 0.3T\). What is the induced EMF in the rod?

**Solution**:

Using the motion EMF formula \(\mathcal{E} = Blv\), we get

\[ \mathcal{E} = 0.3 \cdot 0.5 \cdot 2 = 0.3 V \]

**Problem 5**: A circular loop of wire with a radius of \(0.1m\) is in a magnetic field that is increasing at a rate of \(0.02 T/s\). Find the induced EMF in the loop if it has 5 turns.

**Solution**:

\[ \mathcal{E} = -N\frac{d\Phi_B}{dt} \]

\[ \Phi_B = BA = B\pi r^2 \]

Taking the derivative,

\[ \frac{d\Phi_B}{dt} = \pi r^2 \frac{dB}{dt} \]

Plugging in the values,

\[ \mathcal{E} = -5 \pi (0.1)^2 (0.02) \approx -0.00314 V \]