Maxwell's equation in differential form from Ampere's law is derived as follows:

1. Ampere's law states that the magnetic field in space is proportional to the current that produced it. In integral form, it is expressed as ∮ B . dl = μ0 I, where B is the magnetic field, dl is an infinitesimal element of the path, μ0 is the permeability of free space, and I is the current passing through the loop.

2. To convert this into differential form, we apply Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of the field over any surface bounded by the loop. This gives us ∮ B . dl = ∬ (curl B) . dA.

3. Setting these two equations equal to each other, we get ∬ (curl B) . dA = μ0 I.

4. The current I can be expressed as the surface integral of the current density J, i.e., I = ∬ J . dA.

5. Substituting this into the equation, we get ∬ (curl B) . dA = μ0 ∬ J . dA.

6. Since this equation must hold for all surfaces, the integrands themselves must be equal. This gives us the differential form of Ampere's law: curl B = μ0 J.

7. However, this form of Ampere's law does not include the displacement current term that Maxwell added to account for changing electric fields. Including this term, the final form of Maxwell's equation from Ampere's law is curl B = μ0 (J + ε0 dE/dt), where ε0 is the permittivity of free space, E is the electric field, and dE/dt is the rate of change of the electric field.

Question

Maxwell's equation in differential form from Ampere's law is derived as follows:

1. Ampere's law states that the magnetic field in space is proportional to the current that produced it. In integral form, it is expressed as ∮ B . dl = μ0 I, where B is the magnetic field, dl is an infinitesimal element of the path, μ0 is the permeability of free space, and I is the current passing through the loop.

2. To convert this into differential form, we apply Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of the field over any surface bounded by the loop. This gives us ∮ B . dl = ∬ (curl B) . dA.

3. Setting these two equations equal to each other, we get ∬ (curl B) . dA = μ0 I.

4. The current I can be expressed as the surface integral of the current density J, i.e., I = ∬ J . dA.

5. Substituting this into the equation, we get ∬ (curl B) . dA = μ0 ∬ J . dA.

6. Since this equation must hold for all surfaces, the integrands themselves must be equal. This gives us the differential form of Ampere's law: curl B = μ0 J.

7. However, this form of Ampere's law does not include the displacement current term that Maxwell added to account for changing electric fields. Including this term, the final form of Maxwell's equation from Ampere's law is curl B = μ0 (J + ε0 dE/dt), where ε0 is the permittivity of free space, E is the electric field, and dE/dt is the rate of change of the electric field.

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