Help me distinguish between an Amperian loop and a surface

This image is from the Resnick-Halliday-Krane textbook. These images have been used to see that a correction was needed in Ampere's circuital law.

What I am confused about is, how are the two end surfaces of a 3D surface used as Amperian loops and how do the different results from application of Ampere's law present any anomaly? To me, the 2D surface inside the capacitor should be independent of any result that comes from the 2D surface outside the capacitor.

• It would help to upload just the image and have the image upright. – Aaron Stevens Jul 7 at 15:09
• The image in my gallery was upright but it was uploaded in landscape. Don't know why. – Aabesh Ghosh Jul 7 at 15:43

1 Answer

By Stoke's theorem you are free to use any surface bounded by your Amperian loop. That surface does not need to be contained in a single plane.

The problem that then arises is that Ampere's law as just $$\int\mathbf B\cdot\text d\mathbf l=\mu_0 I_{enc}$$ doesn't work here because the right hand side for each case is different for the two surfaces ($$I_{enc}=i$$ for the top, and $$I_{enc}=0$$ for the bottom), but the left hand side is the same for each case (same field and same Amperian loop).

This is why you need the displacement current. To make the $$I_{enc}$$ the same in both cases so that Ampere's law (Stoke's theorem) is valid.

• I still do not get what you mean by saying that I may take any surface bounded by the Amperian loop. What is the Amperian loop in this example? Can you please elaborate what is the surface that is being considered? – Aabesh Ghosh Jul 7 at 15:42
• @AabeshGhosh It tells you what the loop and surface is in the picture you posted. – Aaron Stevens Jul 7 at 15:46
• Okay so the surface may be 2D/3D, but the loop must enclose it. Okay – Aabesh Ghosh Jul 7 at 15:52