# 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. Jul 7, 2019 at 15:09
• The image in my gallery was upright but it was uploaded in landscape. Don't know why. Jul 7, 2019 at 15:43

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.