E field of continuous planar charge

Electric field of a continuous plane of charge



We can use our previous result of the electric field of a thin ring of charge to derive the E field of a disk.


E field of a disk

We have an expression for the E field from a ring of charge at t a point P along the z-axis. Summing over all of the rings in a disk will give us the total E field from the disk. The rings have radii ri and radius Δr.

We define a surface charge density σ as the total charge divided by the surface area of the disk.



surface charge density

Since we have constant surface charge density, it is the same for each incremental charge per area.



disk integral

We use this expression for the charge density to rewrite our sum in terms of a geometric quantity. This allows us to write it in integral form and then integrate over the disk geometry.



disk integral

We solve the integral using substitution, and evaluate it over the limits of the disk.



disk integral

Factoring z out allows us to get an expression as R goes to infinity. This is a very interesting case, and we will use it to define the electric field of an infinite plane of charge. Notice this expression has no dependence on distance at all. The electric field of an infinite sheet of charge does not diminish with distance.



capacitor

We will use this result to model the electric field inside a capacitor. A parallel plate capacitor consists of two plates with equal and opposite charge. The fact that the size of area of the plates is much larger than their separation allows us to model a parallel plate capacitor as two infinite planes of charge, with a constant electric field between them. The field strength is the superposition of the two fields.



Motion of a charged particle in a constant E field



The constant electric field of an infinite plane of charge allows us to calculate the motion of a charged particle in a manner analogous to the motion of a massive particle in a constant gravity field.

acceleration defined

We use the fact that the net force equals the mass times acceleartion of a particle to define acceleration of a charged particle in a constant electric field.



Sample problem



Consider an infinite charged plate with a surface charge density of 2.0 x 10-6C/m2. An electron is shot upward from the plane at 2.0 x 106 m/s. What is the maximum height above the plane reached by the electron?