Faraday's law of induction

Faraday's Law of Induction, formulated by Michael Faraday in 1831, is a fundamental principle of electromagnetism that describes how a change in magnetic field within a closed loop induces an electromotive force (EMF) in a conductor. This law forms the backbone of many modern technologies, including electric generators, transformers, and inductive charging devices. In essence, Faraday's Law reveals how electricity can be generated from magnetism, which is pivotal in converting mechanical energy into electrical energy and vice versa. In real life, this principle is vividly demonstrated in the working of power plants where mechanical energy from turbines is transformed into electricity, powering our homes and industries. Additionally, it's the principle behind induction stovetops and wireless charging pads for smartphones, showcasing its ubiquitous presence in everyday life. In this article, we will discuss the concept of Faraday's law of induction. Knowing this concept is vital both in theoretical questions and also in its practical applications in the examinations.

What is Faraday’s First Law?

Faraday's First Law of Electromagnetic Induction states that an electromotive force (EMF) is induced in a conductor when the magnetic field around it changes. In simpler terms, it means that whenever a conductor experiences a change in the magnetic environment, a voltage is generated within the conductor. This can occur due to the relative motion between the conductor and the magnetic field, or due to a change in the strength or direction of the magnetic field.

Whenever the number of magnetic lines of force (Magnetic Flux) passing through a circuit changes an emf called induced emf is produced in the circuit. The induced emf persists only as long as there is a change of flux.

What is Faraday’s Second Law?

Faraday's Second Law of Electromagnetic Induction states that the magnitude of the induced electromotive force (EMF) is directly proportional to the rate of change of the magnetic flux through the circuit.

The induced emf is given by the rate of change of magnetic flux linked with the circuit.

i.e Rate of change of magnetic Flux $\varepsilon=\frac{-d \phi}{d t}$

where $d \phi \rightarrow \phi_2-\phi_1=$ change in flux

And For N turns it is given as $\varepsilon=\frac{-N d \phi}{d t}$ where N= Number of turns in the Coil.

The negative sign indicates that induced emf (e) opposes the change of flux.
And this Flux may change with time in several ways

$
\text { 1.e As } \phi=B A \cos \Theta \text { so } \varepsilon=N \frac{-d}{d t}(B A \cos \Theta)
$
1. If Area (A) changes then
$
\varepsilon=-N B \cos \Theta\left(\frac{d A}{d t}\right)
$
2. If Magnetic field (B) changes then
$
\varepsilon=-N A \cos \Theta\left(\frac{d B}{d t}\right)
$
3. If Angle ( $\theta$ ) change then $\varepsilon=-N A B \frac{d(\cos \Theta)}{d \Theta} \times \frac{d \Theta}{d t}$ or $\varepsilon=+N B A \omega \sin \Theta$

Induced Current

Induced current refers to the electric current that is generated in a conductor when it experiences a change in the magnetic field around it. This phenomenon is a direct consequence of Faraday's Law of Induction, which states that a changing magnetic flux through a conductor induces an electromotive force (EMF) that drives the current.

$I=\frac{\varepsilon}{R}=\frac{-N}{R} \frac{d \phi}{d t}$

where

$R \rightarrow$ Resistance
$\frac{d \phi}{d t} \rightarrow_{\text {Rate of change of flux }}$

Induced Charge

Induced charge refers to the redistribution of electric charge within a conductor when it is exposed to an external electric field. Unlike induced current, which involves the flow of electric charge, induced charge involves the movement and separation of charges within a conductor without necessarily creating a continuous current.

$\begin{aligned} & d q=i \cdot d t=\frac{-N}{R} \frac{d \phi}{d t} \cdot d t \\ & d q=\frac{-N}{R} d \phi\end{aligned}$

I.e Induced Charge was time-independent.

Induced Power

Induced power refers to the electrical power generated as a result of electromagnetic induction. This concept is central to many electrical devices and systems, where a changing magnetic field induces an electromotive force (EMF) in a conductor, leading to the generation of electrical power.

$P=\frac{\varepsilon^2}{R}=\frac{N^2}{R}\left(\frac{d \phi}{d t}\right)^2$

i.e. - Induced Power depends on both time and resistance.

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Solved Examples Based on Faraday's Law of Induction

Example 1: The flux linked with a coil at any instant t is given by $\phi=10 t^2-50 t+250$ The induced emf (in Volts) at t = 3s is

1) -10
2) -190
3) 190
4) 10

Solution:

Rate of change of magnetic Flux

$
\varepsilon=\frac{-d \phi}{d t}
$
wherein
$
d \phi \rightarrow \phi_2-\phi_1
$
$
\begin{aligned}
& \phi=10 \mathrm{t}^2-50 t+250 \\
& \therefore \frac{d \phi}{d t}=20 t-50
\end{aligned}
$

Induced em $f, \varepsilon=\frac{-d \phi}{d t}$ or $\varepsilon=-(20 t-50)=-[(20 \times 3)-50]=-10$ volt or $\varepsilon=-10$ volt

Hence, the answer is the option (1).

Example 2: Figure shows three regions of the magnetic field, each of area A, and in each region magnitude of the magnetic field decreases at a constant rate a. If $\vec{E}$ is induced electric field then the value of the line integral $\oint \vec{E}$.$d \vec{r}$ along the given loop is equal to

1) $\alpha A$
2) $-\alpha A$
3) $3 \alpha \mathrm{A}$
4) $-3 \alpha A$

Solution:

Rate of change of magnetic Flux
$
\varepsilon=\frac{-d \phi}{d t}
$
wherein
$
d \phi \rightarrow \phi_2-\phi_1
$
$\phi_2-\phi_1-$ change in flux

Potential
$
\int \vec{E} \cdot d \vec{r}=-\frac{d \phi}{d t}
$
and take the sign of flux according to the right-hand curl rule.
$
\int \vec{E} \cdot d \vec{r}=-((\alpha A)+(\alpha A)+(-\alpha A))=-\alpha A
$

Hence, the answer is the option (2).

Example 3: A coil having n turns and resistance R is connected with a galvanometer of resistance 4R. This combination is moved in time t seconds from a magnetic field W₁ weber to W₂ weber. The induced current in the circuit is

1) $-\frac{W_2-W_1}{5 R n t}$
2) $-\frac{n\left(W_2-W_1\right)}{5 R t}$
3) $-\frac{\left(W_2-W_1\right)}{R n t}$
4) $-\frac{n\left(W_2-W_1\right)}{R t}$

Solution:

$\phi=W=$ flux $\times$ per unit turn of the coil
Change in flux $=W_2-W_1$
Total current per coil
$
\begin{aligned}
& \therefore I=\frac{\xi}{R_{e q}}=\frac{n}{R_{e q}} \frac{\Delta \phi}{\Delta t} \\
& I=\frac{n\left(W_2-W_1\right)}{(R+4 R) t}=\frac{n\left(W_2-W_1\right)}{5 R t}
\end{aligned}
$

The induced current is opposite to its cause of production
$
I=\frac{-n\left(W_2-W_1\right)}{5 R t}
$

Hence, the answer is the option (2).

Example 4: Faraday's law of electromagnetic induction states that the induced EMF is

1) Proportional to the change in magnetic flux linkage

2) Equal to the change in magnetic flux linkage

3) Equal to the change in magnetic flux

4) Proportional to the rate of change of magnetic flux

Solution:

Flux may change with time in several ways

$
\varepsilon=N \frac{-d}{d t}(B A \cos \Theta)
$

From Faraday's law
$
\varepsilon=-N \frac{d \phi}{d t}
$

Where $\phi=B A \cos \theta$

Hence, the answer is the option (4).

Example 5: A small circular loop of wire of radius a is located at the centre of a much larger circular wire loop of radius b. The two loops are in the same plane. The outer loop of radius b carries an alternating current $I=I_0 \cos (\omega t)$. The emf induced in the smaller inner loop is nearly :

1) $\frac{\pi \mu_0 I_0}{2} * \frac{a^2}{b} \omega \sin \omega t$
2) $\frac{\pi \mu_0 I_0}{2} * \frac{a^2}{b} \omega \cos \omega t$
3) $\pi \mu_0 I_0 * \frac{a^2}{b} \omega \sin \omega t$
4) $\pi \mu_0 I_0 * \frac{b^2}{a} \omega \cos \omega t$

Solution:

The magnetic field produced by the outer loop $=\frac{\mu_o I}{2 R}=\frac{\mu_o I_o \cos \omega t}{2 b}$

$\begin{aligned} & \phi=B \cdot A=\left(\frac{\mu_o I_o \cos \omega t}{2 b}\right) \pi a^2 \\ & \xi=\left|\frac{-d \phi}{d t}\right|=\frac{\mu_o I_o \pi}{2 b} a^2 \cdot \omega \sin \omega t\end{aligned}$

Hence the answer is the option (1).

Summary

Faraday's Law of Induction, established by Michael Faraday, describes how a changing magnetic field induces an electromotive force (EMF) in a conductor. Faraday's First Law states that EMF is generated when magnetic flux through a circuit changes, while the Second Law quantifies that the magnitude of EMF is proportional to the rate of flux change. Induced currents and charges arise from these EMFs, leading to practical applications in power generation, transformers, and wireless charging. Understanding these principles is crucial for both theoretical analysis and practical problem-solving in electromagnetism.