The Equals Sign in Ampere's Law
The Equals Sign (=)
The Equals Sign (=) in Equation 4, or Ampere's Law is actually different from the equals signs in the first two (scalar) Maxwell's Equations but similar to the equal's sign in Faraday's Law. In Ampere's Law, the equals sign relates two 3-D vector fields on either side. Therefore, this equals sign can be regarded as 3 scalar equals signs. To see this clearly, let's rewrite Ampere's law as F = G, where F and G are two 3-dimensional vector fields:
Then the true meaning of the = sign in Ampere's Law can be shown in Equation [2] - it means the x-component of the F vector must equal the x-component of the G vector, the y-components must be equal, and teh z-components are equal:
We see from [2] that the vector equals = is equivalent to 3 scalar equals signs. Therefore, while it seems trivial to define the equality signs in Maxwell's Equations, they do have different meanings. The final note about this equals: it relates quantities with units of Amperes per meter squared [A/m^2]. It is important to know the units of what you are measuring, so this is not info that should be discarded.
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