Current in an infinite current sheet

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In the discussion on an infinite current sheet, the application of Ampere’s law reveals that the enclosed current is I = σL, where σ is the surface current density. This raises a question about the units, as it appears to suggest that the right-hand side has units of charge per meter (Q/m). However, it is clarified that the correct interpretation involves the relationship I = dQ/dt = σ(dA/dt), emphasizing that surface current density (A/m) multiplied by length (m) results in current (A). The focus is on ensuring proper unit consistency in the context of magnetic field calculations. Understanding these relationships is crucial for accurately applying Ampere’s law to infinite current sheets.
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Suppose we have a infinite current sheet of surface density \sigma and apply Ampere’s law to find the magnetic field. Using a rectangular loop of side lengths L, why would the enclosed current be I=\sigma L? Doesn't this imply the RHS has units of \frac{Q}{m}?

Shouldn't we be looking at this: I=\frac{dQ}{dt}=\sigma \frac{dA}{dt}?
 
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A surface current density has units of A/m in the SI system. Multiplying that by length (units of m) gives current (units of A).
 
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