Faraday's Law in Differential Form
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You should be comfortable with the integral form of Faraday’s law and the definition of magnetic flux. Familiarity with the vector calculus identity relating the circulation of a field to its curl over a surface is essential.
Why this matters
The differential form of Faraday’s law is the local statement that a changing magnetic field creates a circulating electric field.
In the previous lecture, we treated induction as a global phenomenon affecting an entire circuit. While useful for calculating EMF in specific geometries, this approach obscures the fundamental local physics. By moving to the differential form, we isolate the source of the electric field at every point in space. This local perspective is not merely a mathematical convenience; it is a prerequisite for deriving Maxwell’s equations in their final form, understanding electromagnetic wave propagation, and establishing the relativistic covariance of the electromagnetic field. Without the differential form, we cannot describe how fields evolve in vacuum or how energy is transported by waves.
The Local Source of Electric Circulation
Imagine a region of space where the magnetic field is changing in time. In the integral view, we see a loop of wire and a changing flux, concluding that a current flows. But what if there were no wire? Does the electric field still exist? The differential form answers this by stating that the electric field itself possesses a non-zero curl wherever the magnetic field changes. This curl represents the "swirl" or circulation of the electric field lines. Unlike electrostatic fields, which are conservative and have zero curl, the induced electric field is non-conservative. Its field lines form closed loops, not starting or ending on charges. This is a profound shift: the electric field is no longer solely a response to charge density; it is also a direct response to the time-varying magnetic field. The relationship is local and instantaneous at each point, independent of any material circuit.
From Integral to Differential: Stokes' Theorem
To transition from the global flux rule to the local law, we apply Stokes' theorem. This 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 \( C \). Mathematically, this is expressed as:
We equate this to the integral form of Faraday’s law, which states that the line integral of the electric field is equal to the negative rate of change of the magnetic flux through the surface. The magnetic flux is defined as the surface integral of the magnetic field \( \mathbf{B} \). Assuming the surface \( S \) is stationary in space (so that the time derivative can be moved inside the integral), we have:
Since this equality holds for any arbitrary surface \( S \), the integrands themselves must be equal at every point in space. This leads directly to the differential form of Faraday’s law:
This equation is one of the four Maxwell equations. It asserts that the curl of the electric field is determined entirely by the time derivative of the magnetic field. The negative sign is a consequence of Lenz’s law, ensuring that the induced field opposes the change in flux that created it.
Physical Interpretation: Non-Conservative Fields
The most significant implication of \( \nabla \times \mathbf{E} \neq 0 \) is that the electric field is no longer conservative. In electrostatics, the work done in moving a charge around a closed loop is zero. Here, the work done is non-zero and equal to the negative rate of change of magnetic flux. This means that the electric potential \( V \) cannot be uniquely defined in the presence of a time-varying magnetic field. The concept of voltage becomes path-dependent. If you measure the potential difference between two points in a circuit with a changing magnetic field, the reading depends on the path taken by the measuring instrument. This is why we speak of "EMF" (electromotive force) rather than "potential difference" in induction problems. The field lines of \( \mathbf{E} \) form closed loops, circulating around the region of changing \( \mathbf{B} \). This is a fundamentally different topology from the radial field lines of a static charge distribution.
Consistency with Gauss's Law for Magnetism
It is worth checking that this law is consistent with the other field equations. Taking the divergence of both sides of the differential Faraday’s law, we use the vector identity that the divergence of a curl is always zero:
Therefore, the right-hand side must also have zero divergence:
This is consistent with Gauss’s law for magnetism, \( \nabla \cdot \mathbf{B} = 0 \), which states that there are no magnetic monopoles. The time-varying magnetic field does not create or destroy charge; it merely redistributes the electric field. So Faraday's law guarantees that if \( \nabla \cdot \mathbf{B} = 0 \) holds at one instant it holds for ever: the two equations are compatible. (Charge conservation is the corresponding check for the Ampère–Maxwell law, met in lecture 5.)
The derivation
Faraday's Law and the Flux Rule
For any circuit C bounding a surface S, the electromotive force equals minus the rate of change of magnetic flux, ℰ = −dΦB/dt, and this single "flux rule" resolves into two physically distinct contributions — the induced curl of the electric field, giving the differential law ∇ × E = −∂B/∂t, and the motional term ∮ (v × B)·dl arising from the Lorentz force on charges carried by a moving or deforming conductor.
A subtle point: \(\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t\) holds for the total electric field, not just an "induced part". Splitting \(\mathbf{E}\) into a Coulomb part (zero curl) and an induced part is often useful, but the split is not unique, and with potentials it is gauge-dependent: \(\mathbf{E} = -\nabla\phi - \partial\mathbf{A}/\partial t\). What is unambiguous is that whenever \(\mathbf{B}\) changes, \(\mathbf{E}\) has a non-zero curl somewhere, so no single-valued scalar potential can describe it. C
Worked example
Problem. A long solenoid of radius \(a = 0.050\ \mathrm{m}\) with \(n = 1000\) turns per metre carries a current rising at \(dI/dt = 2.0\ \mathrm{A\,s^{-1}}\). Find the induced electric field at \(r = 0.020\ \mathrm{m}\) (inside) and at \(r = 0.10\ \mathrm{m}\) (outside).
- Inside, \(B = \mu_0 n I\), so \(\partial B/\partial t = \mu_0 n\, dI/dt = 4\pi \times 10^{-7} \times 1000 \times 2.0 = 2.51 \times 10^{-3}\ \mathrm{T\,s^{-1}}\).
- By symmetry \(\mathbf{E}\) circles the axis with constant magnitude on a circle of radius \(r\); Faraday's law gives \(E\,(2\pi r) = \pi r^2\, \partial B/\partial t\) for \(r < a\).
- So \(E = \tfrac{r}{2}\,\partial B/\partial t = 0.010 \times 2.51 \times 10^{-3} = 2.51 \times 10^{-5}\ \mathrm{V\,m^{-1}}\) at \(r = 0.020\ \mathrm{m}\).
- Outside (\(r > a\)) the flux is confined to \(\pi a^2\): \(E\,(2\pi r) = \pi a^2\, \partial B/\partial t\), so \(E = \dfrac{a^2}{2r}\,\partial B/\partial t = \dfrac{(0.050)^2}{0.20} \times 2.51 \times 10^{-3} = 3.14 \times 10^{-5}\ \mathrm{V\,m^{-1}}\) at \(r = 0.10\ \mathrm{m}\).
- The field outside is non-zero even though \(B = 0\) there: the curl of \(\mathbf{E}\) vanishes outside, but its circulation around the solenoid does not.
Units check. \(\mathrm{m} \times \mathrm{T\,s^{-1}} = \mathrm{m} \times \mathrm{V\,s\,m^{-2}\,s^{-1}} = \mathrm{V\,m^{-1}}\).
Three quick tests
- What is the value of \( \nabla \cdot \mathbf{E} \) in a region where \( \rho = 0 \) but \( \mathbf{B} \) is changing?
Answer
Zero. Gauss's law for electricity states \( \nabla \cdot \mathbf{E} = \rho/\epsilon_0 \). The time-varying \( \mathbf{B} \) affects the curl of \( \mathbf{E} \), not its divergence. - Can the electric field be conservative if \( \partial \mathbf{B}/\partial t \neq 0 \)?
Answer
No. A conservative field must have zero curl. Faraday's law shows that a changing \( \mathbf{B} \) produces a non-zero curl in \( \mathbf{E} \). - How does the induced electric field behave outside a long solenoid where \( B=0 \)?
Answer
It is non-zero. Although \( B=0 \) outside, the flux through a loop outside the solenoid is non-zero (equal to the flux inside). Thus, \( \oint \mathbf{E} \cdot d\mathbf{l} \neq 0 \), implying a circulating electric field outside.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.