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Question

Explain the contradiction in Ampere's Law.


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Solution

Ampere's Law

Ampere's circuital law states that the line integral of the magnetic field forming a closed loop around the current ( I ) carrying wire in the plane normal to the current is equal to 𝜇otimes the net current traveling through the closed-loop.

Ampere's law (original) - B.dl=𝜇o𝐼

Maxwell Ampere's law

  1. Ampere's circuital law expresses the relationship between circuit current and magnetic field line integral.
  2. At the time when Ampere gave this law, it appeared to be valid based on human knowledge at the time. It was only after Maxwell understood the exception to Ampere's law that he realized it was invalid.
  3. There was some discrepancy in Ampere's circuital law after Maxwell conceived of a circumstance in which the law was not valid.
  4. Maxwell corrected this discrepancy by adding new terms to the original Ampere's equation, such as displacement current, and the law has been known as Maxwell Ampere's law ever since.

Maxwell Ampere's law - B.dl=μoi+μoid

Displacement current

  1. Displacement current occurs whenever the electric field (and thus the electric flux) in a region changes, resulting in a unique sort of current known as displacement current.
  2. It is proportional to the rate of change of electric flux in the region mathematically. i.e. id=μodϕEdt

Proving contradiction in Ampere's Law

  1. ‘i’ is the current enclosed by the loop. In a normal full circuit, the current enclosed is the actual current flowing in the circuit, and the electric field cannot vary if the battery voltage remains constant.
  2. As a result, in that scenario, the displacement current is zero. In these types of situations, the law is applicable.
  3. However, in a capacitive circuit, if we take the loop (see loop2 in the diagram) encompassing half of the capacitor, the current enclosed in the circuit is zero because no current passes through the capacitor (Note that it flows through the circuit only till capacitor charge fully, after that it stops flowing in the circuit too). As a result, in that scenario, i=0.
  4. However, both loops can be used, and the laws must be valid. This was the regularity.
  5. When a battery is connected to a capacitor, the electric flux associated with it varies, resulting in the generation of displacement current.
  6. Hence for the second loop, the law becomes B.dl=μoid

Hence, we have proved the contradiction in Ampere's law.


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