Physics · Electromagnetic Waves · NEET
In vacuum the four integral-form equations are: 1) Closed-surface E-flux: oint E . dA = Q/e0 (Gauss's law for electricity). 2) Closed-surface B-flux: oint B . dA = 0 (Gauss's law for magnetism). 3) Closed-loop E: oint E . dl = -(dPhiB/dt) (Faraday's law). 4) Closed-loop B: oint B . dl = mu0 ic + mu0 e0 (dPhiE/dt) (Ampere-Maxwell law). The first two use a closed surface (flux). The last two use a closed loop (line integral). Getting surface vs loop right is a common NEET slip.
Only the fourth, the Ampere-Maxwell law: oint B . dl = mu0 ic + mu0 e0 (dPhiE/dt). The extra term mu0 e0 (dPhiE/dt) is Maxwell's displacement current term. It says a changing electric flux also makes a magnetic field, just like a real conduction current ic does. This is the term Maxwell added to fix Ampere's original law.
oint B . dA = 0 means the net magnetic flux out of any closed surface is always zero. Physically this says there are no magnetic monopoles (no isolated north or south pole). Magnetic field lines always form closed loops, so every line that enters a closed surface must also leave it. Compare with electricity, where oint E . dA = Q/e0 is non-zero because isolated electric charges (Q) do exist.
Faraday's law (3) says a changing magnetic field creates an electric field. The Ampere-Maxwell law (4) says a changing electric field creates a magnetic field. So the two changing fields keep regenerating each other and travel outward together as a wave. Solving equations 3 and 4 in empty space gives a wave moving at speed c = 1/sqrt(mu0 e0), which matches the measured speed of light.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The vacuum forms above use mu0 and e0 (free-space constants) with only real charge Q and conduction current ic as sources. In a material medium you replace e0 by e and mu0 by mu, which changes the wave speed to v = 1/sqrt(mu e). The structure of the four equations stays the same.
James Clerk Maxwell. Ampere's original circuital law failed for a charging capacitor because no conduction current flows in the gap between the plates. Maxwell added the mu0 e0 (dPhiE/dt) term so the law works everywhere, which completed the set of four equations.
Maxwell did not invent all four from scratch. Gauss's law, Gauss's law for magnetism, and Faraday's law were already known. Maxwell's key contribution was fixing Ampere's law by adding the displacement current term, and then unifying all four into one consistent set that predicted light as an electromagnetic wave.
The existence of electromagnetic waves travelling at speed c = 1/sqrt(mu0 e0) = 3 x 10^8 m/s in vacuum. Because this equals the measured speed of light, Maxwell concluded that light itself is an electromagnetic wave, unifying electricity, magnetism, and optics.
