Faraday's Law: Integral and Differential Forms

Hello! Welcome to your next lesson in Electrodynamics.

In our last session, we deeply investigated the Lorentz force, analyzing how static electric and magnetic fields dictate the motion of individual charged particles. We saw that the magnetic force, $F_{B} = q (v \times B)$ , is responsible for motional EMF in a moving conductor. This raised a profound question, one that puzzled Einstein himself: what if the conductor is stationary, but the magnetic field itself changes?

Today, we answer that question by exploring one of the four pillars of classical electromagnetism: Faraday's Law of Induction. This law reveals a fundamental connection between electricity and magnetism, showing that a changing magnetic field creates an electric field.

Your learning outcome is to derive Faraday's Law of Induction in integral and differential forms ( $\nabla \times E = - \partial B / \partial t$ ). This is a critical step in assembling the complete set of Maxwell's Equations and has immense practical importance, forming the basis for electric generators, transformers, and induction motors.

1. The Phenomenon of Induction

Michael Faraday's experiments in the 1830s demonstrated that a changing magnetic environment could induce a current in a wire loop, even without a battery. This "electromotive force" (EMF) could be generated by:

Moving a permanent magnet near the loop.
Moving the loop near a permanent magnet.
Changing the current in a nearby electromagnet.

The common factor in all these cases is a change in magnetic flux through the loop.

Magnetic Flux and Electromotive Force (EMF)

To formalize this, we need two key definitions from vector calculus:

Magnetic Flux ( $Φ_{B}$ ): The amount of magnetic field passing through a surface $S$ . It's calculated by the surface integral of the magnetic field $B$ .
$Φ_{B} = \int_{S} B \cdot d S$
where $d S$ is an infinitesimal area vector perpendicular to the surface.
Electromotive Force ( $E$ ): The work done per unit charge in moving a charge around a closed loop $C$ . It is given by the closed line integral of the electric field $E$ .
$E = \oint_{C} E \cdot d l$
For the electrostatic fields we studied earlier (caused by static charges), this integral is always zero, signifying a conservative field. As we'll see, the electric field induced by a changing magnetic field is non-conservative.

Faraday's discovery, in modern mathematical language, is that the induced EMF is equal to the negative rate of change of magnetic flux.

2. Faraday's Law: The Integral Form

The relationship Faraday discovered is now known as Faraday's Law of Induction in its integral form. It is the third of Maxwell's four equations.

\oint_{C} E \cdot d l = - \frac{d Φ _{B}}{d t} = - \frac{d}{d t} \int_{S} B \cdot d S

This equation is a complete statement of the law at a macroscopic level. It relates the total EMF around a loop $C$ to the rate of change of the total magnetic flux passing through the surface $S$ bounded by that loop.

The negative sign is crucial and represents Lenz's Law. It signifies that the induced current will flow in a direction that creates its own magnetic field to oppose the change in magnetic flux that produced it. This opposition is a direct consequence of the conservation of energy.

The following video provides an excellent derivation and explanation of the concepts we've discussed so far.

Maxwell-Faraday Equation (Faraday's Law of Induction) in Integral and Differential Form

The video 'Maxwell-Faraday Equation...' from Elucyda gives a clear and mathematically sound walkthrough of Faraday's Law. Let's start with its integral form.

Please watch from the beginning to 8:45. This part introduces the integral form of Faraday's Law, carefully defining each term (EMF, magnetic flux, line and surface integrals) and provides an intuitive explanation of what the law means, including the role of Lenz's Law.

3. Deriving the Differential Form

The integral form is excellent for analyzing circuits and loops, but to understand what's happening at a specific point in space, we need a local, differential form. The derivation is a beautiful application of vector calculus, specifically Stokes' Theorem, which you've encountered in your prerequisite math module.

Our starting point is the integral form:

\oint_{C} E \cdot d l = - \frac{d}{d t} \int_{S} B \cdot d S

We will transform both sides of this equation to be surface integrals over the same surface $S$ .

Step 1: Transform the Left-Hand Side using Stokes' Theorem

Stokes' Theorem states that the line integral of a vector field around a closed curve $C$ is equal to the surface integral of the curl of that field over any surface $S$ bounded by $C$ .

\oint_{C} E \cdot d l = \int_{S} (\nabla \times E) \cdot d S

This theorem is the fundamental link between the macroscopic (circuital) and microscopic (differential) forms of the law.

Step 2: Manipulate the Right-Hand Side

We can move the time derivative inside the integral. Since we are considering a fixed surface in space, the total time derivative $d / d t$ becomes a partial derivative $\partial / \partial t$ acting on the magnetic field $B$ , which may vary in time at each point in space.

- \frac{d}{d t} \int_{S} B \cdot d S = - \int_{S} \frac{\partial B}{\partial t} \cdot d S

Step 3: Equate and Conclude

Now, we substitute both transformed sides back into the equation:

\int_{S} (\nabla \times E) \cdot d S = \int_{S} (- \frac{\partial B}{\partial t}) \cdot d S

This can be rewritten as:

\int_{S} ((\nabla \times E) + \frac{\partial B}{\partial t}) \cdot d S = 0

This equation must hold true for any choice of surface $S$ . The only way a non-zero integrand could integrate to zero over every possible surface is if the integrand itself is identically zero at every point.

Therefore, we arrive at the differential form of Faraday's Law:

\nabla \times E = - \frac{\partial B}{\partial t}

This elegant equation is a local statement: the "curliness" or circulation of the electric field at a point is determined by the rate of change of the magnetic field at that same point. This induced E-field is non-conservative because its curl is non-zero.

To see this derivation in action, you can use the following resources. The video walks through the steps dynamically, while the PDF provides a concise reference.

Maxwell-Faraday Equation (Faraday's Law of Induction) in Integral and Differential Form

Let's continue with the Elucyda video, which now performs the exact derivation we just outlined.

Watch the segment from 8:45 to 14:58. This section rigorously applies Stokes' Theorem to derive the differential form from the integral form, just as we did above.

Chapter 16: Faradays' Law of Induction

For a clear, written summary of this derivation, refer to this document from Valpo Scholar.

Please read page 10 of the PDF, starting from the section 'Final Form of Faraday’s Law of Induction'. It presents the derivation compactly from Equation 16.19 to Equation 16.24. This is a great static reference for the steps involved.

Test your understanding!

In electrostatics, we learned that $\nabla \times E = 0$ . Today, we derived $\nabla \times E = - \frac{\partial B}{\partial t}$ . Does this mean the electrostatic law is wrong? Explain the apparent contradiction.

Show answer

There is no contradiction; the laws apply to different physical situations.

$\nabla \times E = 0$ is the law for electrostatics, which describes fields created by static (stationary) charges. In this case, there are no changing magnetic fields, so $\frac{\partial B}{\partial t} = 0$ , and Faraday's law simplifies to the electrostatic law. This E-field is conservative.
$\nabla \times E = - \frac{\partial B}{\partial t}$ is the more general law, part of electrodynamics. It states that an electric field can be produced by two sources: static charges (the conservative part) and changing magnetic fields (the non-conservative, induced part). The total electric field is the sum of both contributions.

4. Application and Challenge

The concepts of induced electric fields are central to solving advanced problems in electromagnetism. Your goal to tackle Olympiad and JEE Advanced-level problems requires a deep, intuitive understanding of what $\nabla \times E$ physically represents.

Consider the following problem. You don't need to solve it now, but read it and think about how the principles we've discussed are applied.

Electromagnetic Induction Problem: Rotating Charged Cylinder and Ring — A complex problem from the Pathfinder Solutions Series involving a rotating charged cylinder inducing a torque on a coaxial charged ring.

How today's lesson applies:

The hollow cylinder is charged and rotating. A moving charge is a current. This rotating charge creates a magnetic field $B$ inside the cylinder.
When the cylinder's rotation starts from zero and increases, the magnetic field $B$ inside it changes with time ( $\partial B / \partial t \neq = 0$ ).
According to Faraday's Law, this changing $B$ field induces a curly electric field, $E$ , inside the cylinder ( $\nabla \times E \neq = 0$ ).
This induced electric field is not radial like a Coulomb field; it will have a tangential component that circulates around the axis.
This tangential $E$ field will exert a force ( $F = q E$ ) on the charges of the inner ring, producing a torque that causes the ring to rotate.

Solving this problem requires quantifying each of these steps, but the core physical principle is precisely the differential form of Faraday's Law.

Conclusion

Today we have made a significant leap from static to dynamic fields. We've seen that magnetism isn't just about forces on moving charges; changing magnetism is a source for the electric field itself.

Key Takeaways:

A changing magnetic flux $Φ_{B}$ through a loop induces an electromotive force (EMF) in it.
The Integral Form of Faraday's Law, $\oint E \cdot d l = - d Φ_{B} / d t$ , relates the macroscopic quantities of EMF and flux.
Using Stokes' Theorem, we derived the Differential Form, $\nabla \times E = - \partial B / \partial t$ . This is a local, fundamental equation describing how a time-varying $B$ field generates a spatially-varying (curly) $E$ field.
The electric field induced by a changing magnetic field is non-conservative.

Preview of the Next Lesson:
We have established that changing currents (via changing $B$ fields) can induce EMFs in circuits. In the next lesson, we will quantify this effect by defining mutual and self-inductance. This will allow us to analyze the behavior of inductors as circuit components and explore the crucial concept of energy stored in magnetic fields.