Why does electric field lines bulge out at the endings in a parallel plate capacitor?

Question

I am trying to draw out electric field lines for different conducting bodies, and I know that electric lines in case metallic conductors incident and emerge out perpendicularly, considering a parallel plate capacitor's base component is a metallic conducting plate(s), the electric fields should be perpendicular through out its surface, but in various textbooks they are shown to be bulged out around the edges, why?

Answer 1

By considering the setup at a remote distance, it is perhaps a bit more intuitive why the field lines bulge out. Since the plates are of finite extent and are oppositely charged, at large distance, the field should be asymptotically dipolar. The bulges are the usual bulges of a dipole field which has azimuthal dependance.

Note that the field lines always leave at a normal of the conductors' boundaries. Interpreting the electric field as a velocity field and Gauss' law as a condition of incompressibility, the fringe field lines are expelled from the axis of the capacitor. Indeed, there are already field lines there, so they will need to take a roundabout route so as not to "trespass" this region (forbidden by incompressibility).

Hope this helps.

Answer 2

I will make four points that add up to a justification.

1. Consider the figure shown. We first imagine the electric field being uniform between the plates of the capacitor, and then suddenly stopping at the edge. We know that for an electrostatic situation, $$\oint\vec{E}\cdot\vec{\mathrm{d}\ell}=0.$$

This means that following the red loop, we should get a net integral of zero. But right now, there is a positive contribution along the path from $d$ to $a$, and zero everywhere else. This means that to compensate, we must introduce a field beyond the edge of the capacitor plate.

2. Field lines in electrostatics must begin and end either on a charge or at infinity. So we draw a candidate field line outside the edge of the plates. We perform a red loop again, but for simplicity we adjust the contour of the integral to move along or perpendicular to field lines. We now have a positive contribution from $d$ to $a$, and a negative contribution from $b$ to $c$. But the path from $b$ to $c$ is longer, so if we want a negative contribution of the right size, the field outside must be weaker, to compensate for the longer path. This means that the field lines are more spaced out.

3. We can imagine building the planes of charge by adding one charge at a time, and superposing their individual fields. Each individual charge has field lines heading radially outward. When we have lots of them, and we consider a point near the center of the plates, we have roughly equal contributions pointing in the +X and -X directions, which cancel out, leaving a field only in the -Y direction. But near the edge, there is no such symmetry. So we see these curving field lines with both X and Y components.

4. Finally, as @LPZ said, if we zoom out, we would expect (at some intermediate distance) that the arrangement of the charged plates resemble a pair of opposite point charges - a dipole. And we know what the field of the electric dipole looks like, and there are bulges similar to that.