It seems like your question is incomplete or not clear. However, I can explain how Maxwell's equations in differential form are obtained from Ampere's law.

1. Ampere's law states that the magnetic field in space is proportional to the electric current which is present in that space. Mathematically, it is represented as ∮ B . dl = μ0 I, where B is the magnetic field, dl is the length element, μ0 is the permeability of free space, and I is the current.

2. To convert this into differential form, we use the concept of divergence and curl from vector calculus. The curl of a vector field is a measure of its "rotation" or "circulation".

3. Applying the 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 that loop, we can convert the line integral in Ampere's law into a surface integral.

4. This gives us ∫ (curl B) . dA = μ0 I, where dA is the area element.

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

6. Substituting this into the equation gives us ∫ (curl B) . dA = μ0 ∫ J . dA.

7. For this equation to hold for all surfaces, the integrands must be equal, giving us the differential form of Ampere's law: curl B = μ0 J.

8. However, this form of Ampere's law does not include displacement current which is important when dealing with changing electric fields. Maxwell added this term to give the final form of the equation: curl B = μ0 (J + ε0 dE/dt), where ε0 is the permittivity of free space, E is the electric field, and d/dt denotes the time derivative.

I hope this helps! If you have any other questions, feel free to ask.

Question

It seems like your question is incomplete or not clear. However, I can explain how Maxwell's equations in differential form are obtained from Ampere's law.

1. Ampere's law states that the magnetic field in space is proportional to the electric current which is present in that space. Mathematically, it is represented as ∮ B . dl = μ0 I, where B is the magnetic field, dl is the length element, μ0 is the permeability of free space, and I is the current.

2. To convert this into differential form, we use the concept of divergence and curl from vector calculus. The curl of a vector field is a measure of its "rotation" or "circulation".

3. Applying the 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 that loop, we can convert the line integral in Ampere's law into a surface integral.

4. This gives us ∫ (curl B) . dA = μ0 I, where dA is the area element.

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

6. Substituting this into the equation gives us ∫ (curl B) . dA = μ0 ∫ J . dA.

7. For this equation to hold for all surfaces, the integrands must be equal, giving us the differential form of Ampere's law: curl B = μ0 J.

8. However, this form of Ampere's law does not include displacement current which is important when dealing with changing electric fields. Maxwell added this term to give the final form of the equation: curl B = μ0 (J + ε0 dE/dt), where ε0 is the permittivity of free space, E is the electric field, and d/dt denotes the time derivative.

I hope this helps! If you have any other questions, feel free to ask.

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